chapter 4 resource masters - ktl math...
TRANSCRIPT
Chapter 4Resource Masters
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9
ANSWERS FOR WORKBOOKS The answers for Chapter 4 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-828007-9 Algebra 2Chapter 4 Resource Masters
1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02
Glenc oe/ M cGr aw-H ill
© Glencoe/McGraw-Hill iii Glencoe Algebra 2
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 4-1Study Guide and Intervention . . . . . . . . 169–170Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 171Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Reading to Learn Mathematics . . . . . . . . . . 173Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 174
Lesson 4-2Study Guide and Intervention . . . . . . . . 175–176Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 177Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 178Reading to Learn Mathematics . . . . . . . . . . 179Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 180
Lesson 4-3Study Guide and Intervention . . . . . . . . 181–182Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 183Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Reading to Learn Mathematics . . . . . . . . . . 185Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 186
Lesson 4-4Study Guide and Intervention . . . . . . . . 187–188Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 189Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 190Reading to Learn Mathematics . . . . . . . . . . 191Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 192
Lesson 4-5Study Guide and Intervention . . . . . . . . 193–194Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 195Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Reading to Learn Mathematics . . . . . . . . . . 197Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 198
Lesson 4-6Study Guide and Intervention . . . . . . . . 199–200Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 201Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 202Reading to Learn Mathematics . . . . . . . . . . 203Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 204
Lesson 4-7Study Guide and Intervention . . . . . . . . 205–206Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 207Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Reading to Learn Mathematics . . . . . . . . . . 209Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 210
Lesson 4-8Study Guide and Intervention . . . . . . . . 211–212Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 213Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Reading to Learn Mathematics . . . . . . . . . . 215Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 216
Chapter 4 AssessmentChapter 4 Test, Form 1 . . . . . . . . . . . . 217–218Chapter 4 Test, Form 2A . . . . . . . . . . . 219–220Chapter 4 Test, Form 2B . . . . . . . . . . . 221–222Chapter 4 Test, Form 2C . . . . . . . . . . . 223–224Chapter 4 Test, Form 2D . . . . . . . . . . . 225–226Chapter 4 Test, Form 3 . . . . . . . . . . . . 227–228Chapter 4 Open-Ended Assessment . . . . . . 229Chapter 4 Vocabulary Test/Review . . . . . . . 230Chapter 4 Quizzes 1 & 2 . . . . . . . . . . . . . . . 231Chapter 4 Quizzes 3 & 4 . . . . . . . . . . . . . . . 232Chapter 4 Mid-Chapter Test . . . . . . . . . . . . 233Chapter 4 Cumulative Review . . . . . . . . . . . 234Chapter 4 Standardized Test Practice . . 235–236Unit 1 Test/Review (Ch. 1–4) . . . . . . . . 237–238
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A36
© Glencoe/McGraw-Hill iv Glencoe Algebra 2
Teacher’s Guide to Using theChapter 4 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 4 Resource Masters includes the core materials neededfor Chapter 4. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 4-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.
Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
© Glencoe/McGraw-Hill v Glencoe Algebra 2
Assessment OptionsThe assessment masters in the Chapter 4Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 216–217. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
44
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 4.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
Cramer’s Rule
KRAY·muhrs
determinant
dilation
dy·LAY·shuhn
element
expansion by minors
identity matrix
image
inverse
isometry
eye·SAH·muh·tree
matrix
MAY·trihks
(continued on the next page)
© Glencoe/McGraw-Hill vii Glencoe Algebra 2
© Glencoe/McGraw-Hill viii Glencoe Algebra 2
Vocabulary Term Found on Page Definition/Description/Example
matrix equation
preimage
reflection
rotation
scalar multiplication
SKAY·luhr
square matrix
transformation
translation
vertex matrix
zero matrix
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
44
Study Guide and InterventionIntroduction to Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 169 Glencoe Algebra 2
Less
on
4-1
Organize Data
Matrixa rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets.
A matrix can be described by its dimensions. A matrix with m rows and n columns is an m � n matrix.
Owls’ eggs incubate for 30 days and their fledgling period is also 30 days. Swifts’ eggs incubate for 20 days and their fledgling period is 44 days.Pigeon eggs incubate for 15 days, and their fledgling period is 17 days. Eggs of theking penguin incubate for 53 days, and the fledgling time for a king penguin is360 days. Write a 2 � 4 matrix to organize this information. Source: The Cambridge Factfinder
What are the dimensions of matrix A if A � ?
Since matrix A has 2 rows and 4 columns, the dimensions of A are 2 � 4.
State the dimensions of each matrix.
1. 4 � 4 2. [16 12 0] 1 � 3 3. 5 � 2
4. A travel agent provides for potential travelers the normal high temperatures for themonths of January, April, July, and October for various cities. In Boston these figures are36°, 56°, 82°, and 63°. In Dallas they are 54°, 76°, 97°, and 79°. In Los Angeles they are68°, 72°, 84°, and 79°. In Seattle they are 46°, 58°, 74°, and 60°, and in St. Louis they are38°, 67°, 89°, and 69°. Organize this information in a 4 � 5 matrix. Source: The New York Times Almanac
Boston Dallas Los Angeles Seattle St. LouisJanuary 36 54 68 46 38
April 56 76 72 58 67 July 82 97 84 74 89
October 63 79 79 60 69
71 44 39 27 45 16 92 53 78 65
15 5 27 �4 23 6 0 5 14 70 24 �3 63 3 42 90
13 10 �3 45 2 8 15 80
Owl Swift Pigeon King PenguinIncubation 30 20 15 53
Fledgling 30 44 17 360
Example 1Example 1
Example 2Example 2
ExercisesExercises
© Glencoe/McGraw-Hill 170 Glencoe Algebra 2
Equations Involving Matrices
Equal MatricesTwo matrices are equal if they have the same dimensions and each element of one matrix isequal to the corresponding element of the other matrix.
You can use the definition of equal matrices to solve matrix equations.
Solve � for x and y.
Since the matrices are equal, the corresponding elements are equal. When you write thesentences to show the equality, two linear equations are formed.4x � �2y � 2y � x � 8
This system can be solved using substitution.4x � �2y � 2 First equation
4x � �2(x � 8) � 2 Substitute x � 8 for y.
4x � �2x � 16 � 2 Distributive Property
6x � 18 Add 2x to each side.
x � 3 Divide each side by 6.
To find the value of y, substitute 3 for x in either equation.y � x � 8 Second equation
y � 3 � 8 Substitute 3 for x.
y � �5 Subtract.
The solution is (3, �5).
Solve each equation.
1. [5x 4y] � [20 20] 2. � 3. �
(4, 5) � , �6� (�2, �7)
4. � 5. � 6. �
(4, 2.5) (6, �3) (�5, 8)
7. � 8. � 9. �
� , �8� �� , � � (18, 21)
10. � 11. � 12. �
(2, 5.2) (2.5, 4) (�12.5, �12.5)
0 �25
x � y x � y
17 �8
2x � 3y �4x � 0.5y
7.5 8.0
3x � 1.5 2y � 2.4
1�2
3�4
5�4
9 51
�3x
� � �7y
� �2
x� � 2y
�3 �11
8x � 6y 12x � 4y
18 20 12 �13
8x � y 16x 12 y � 4x
�1 �18
5x � 3y 2x � y
3 12
2x � 3y x � 2y
�1 22
x � 2y 3x � 4y
4�3
4 � 5x y � 5
�2y x
28 � 4y �3x � 2
3x y
�2y � 2 x � 8
4x y
Study Guide and Intervention (continued)
Introduction to Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
ExampleExample
ExercisesExercises
Skills PracticeIntroduction to Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 171 Glencoe Algebra 2
Less
on
4-1
State the dimensions of each matrix.
1. 2 � 3 2. [0 15] 1 � 2
3. 2 � 2 4. 3 � 3
5. 2 � 4 6. 4 � 1
Solve each equation.
7. [5x 3y] � [15 12] (3, 4) 8. [3x � 2] � [7] 3
9. � (�2, 7) 10. [2x �8y z] � [10 16 �1] (5, �2, �1)
11. � (4, 5) 12. � (2, 4)
13. � (�4, 3) 14. � (0, �2)
15. � (1, �1) 16. � (2, 11, 7)
17. � (9, 4, 3) 18. � (1, 4, 0)
19. � (3, 1) 20. � (12, 3) 4y x � 3
x 3y
6y x
2x y � 2
4x � 1 13 4z
5x 4y � 3 8z
9 4y 9
x 16 3z
7 2y � 4 12 28
3x � 1 18 12 4z
3x y � 3
4x � 1 9y � 5
5x � 2 3y � 10
3x � 2 7y � 2
�20 8y
5x 24
10x 32
20 56 � 6y
4 2
8 � x2y � 8
�14 2y
7x 14
�1 �1 �1 �3
9 3 �3 �6 3 4 �4 5
6 1 2 �3 4 5 �2 7 9
3 2 1 8
3 2 4 �1 4 0
© Glencoe/McGraw-Hill 172 Glencoe Algebra 2
State the dimensions of each matrix.
1. [�3 �3 7] 1 � 3 2. 2 � 3 3. 3 � 4
Solve each equation.
4. [4x 42] � [24 6y] (6, 7)
5. [�2x 22 �3z] � [6x �2y 45] (0, �11, �15)
6. � (�6, 7) 7. � (4, 1)
8. � (�3, 2) 9. � (�1, 1)
10. � (�1, �1, 1) 11. � (7, 13)
12. � (3, �2) 13. � (2, �4)
14. TICKET PRICES The table at the right gives ticket prices for a concert. Write a 2 � 3 matrixthat represents the cost of a ticket.
CONSTRUCTION For Exercises 15 and 16, use the following information.During each of the last three weeks, a road-building crew has used three truck-loads of gravel. The table at the right shows theamount of gravel in each load.
15. Write a matrix for the amount of gravel in each load.
or
16. What are the dimensions of the matrix? 3 � 3
40 40 3232 40 2424 32 24
40 32 2440 40 3232 24 24
Week 1 Week 2 Week 3
Load 1 40 tons Load 1 40 tons Load 1 32 tons
Load 2 32 tons Load 2 40 tons Load 2 24 tons
Load 3 24 tons Load 3 32 tons Load 3 24 tons
6 12 188 15 22
Child Student Adult
Cost Purchasedin Advance
$6 $12 $18
Cost Purchasedat the Door
$8 $15 $22
0 �2
2x � y 3x � 2y
�1 y
5x � 8y 3x � 11
y � 8 17
3x y � 4
�5 x � 1 3y 5z � 4
�5 3x � 1 2y � 1 3z � 2
�3x � 21 8y � 5
6x � 12 �3y � 6
�3x �2y � 16
�4x � 3 6y
20 2y � 3
7x � 8 8y � 3
�36 17
6x 2y � 3
�2 2 �2 3 5 16 0 0 4 7 �1 4
5 8 �1 �2 1 8
Practice (Average)
Introduction to Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
Reading to Learn MathematicsIntroduction to Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 173 Glencoe Algebra 2
Less
on
4-1
Pre-Activity How are matrices used to make decisions?
Read the introduction to Lesson 4-1 at the top of page 154 in your textbook.
What is the base price of a Mid-Size SUV?$27,975
Reading the Lesson
1. Give the dimensions of each matrix.
a. 2 � 3 b. [1 4 0 �8 2] 1 � 5
2. Identify each matrix with as many of the following descriptions that apply: row matrix,column matrix, square matrix, zero matrix.
a. [6 5 4 3] row matrix
b. column matrix; zero matrix
c. [0] row matrix; column matrix; square matrix; zero matrix
3. Write a system of equations that you could use to solve the following matrix equation forx, y, and z. (Do not actually solve the system.)
� 3x � �9, x � y � 5, y � z � 6
Helping You Remember
4. Some students have trouble remembering which number comes first in writing thedimensions of a matrix. Think of an easy way to remember this.
Sample answer: Read the matrix from top to bottom, then from left toright. Reading down gives the number of rows, which is written first inthe dimensions of the matrix. Reading across gives the number ofcolumns, which is written second.
�9 5 6
3x x � y y � z
0 0 0
3 2 5 �1 0 6
© Glencoe/McGraw-Hill 174 Glencoe Algebra 2
TessellationsA tessellation is an arangement of polygons covering a plane without any gaps or overlapping. One example of a tessellation is a honeycomb. Three congruent regular hexagons meet at each vertex, and there is no wasted space between cells. Thistessellation is called a regular tessellation since it is formed by congruent regular polygons.
A semi-regular tessellation is a tessellation formed by two or more regular polygons such that the number of sides of the polygons meeting at each vertex is the same.
For example, the tessellation at the left has two regulardodecagons and one equilateral triangle meeting at eachvertex. We can name this tessellation a 3-12-12 for thenumber of sides of each polygon that meet at one vertex.
Name each semi-regular tessellation shown according to the number of sides of the polygons that meet at each vertex.
1. 2.
3-3-3-3-6 4-6-12
An equilateral triangle, two squares, and a regular hexagon can be used to surround a point in two different orders. Continue each pattern to see which is a semi-regular tessellation.
3. 3-4-4-6 4. 3-4-6-4
not semi-regular semi-regular
On another sheet of paper, draw part of each design. Then determine if it is a semi-regular tessellation.
5. 3-3-4-12 6. 3-4-3-12 7. 4-8-8 8. 3-3-3-4-4not semi-regular not semi-regular semi-regular semi-regular
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
Study Guide and InterventionOperations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
© Glencoe/McGraw-Hill 175 Glencoe Algebra 2
Less
on
4-2
Add and Subtract Matrices
Addition of Matrices � �
Subtraction of Matrices � �
Find A � B if A � and B � .
A � B � �
�
�
Find A � B if A � and B � .
A � B � �
� �
Perform the indicated operations. If the matrix does not exist, write impossible.
1. � 2. �
3. � [�6 3 �2] impossible 4. �
5. � 6. �
�14
� ��145�
��
76
� �161�
10 �1 �13 1 9 �14 1 �2 �2
�12� �
23�
�23� ��
12�
�34� �
25�
��12� �
43�
�2 1 7 3 �4 3 �8 5 6
8 0 �6 4 5 �11 �7 3 4
�6 4�2 1 11 2
�11 6 2 �5 4 �7
5 �2 �4 6 7 9
6 �3 2
2 �2 11 3 13 1
�4 3 2 6 9 �4
6 �5 9 �3 4 5
12 4�12 6
�4 3 2 �12
8 7 �10 �6
�6 11 5 �5 16 �1
�2 � 4 8 � (�3) 3 � (�2) �4 � 1 10 � (�6) 7 � 8
4 �3 �2 1 �6 8
�2 8 3 �4 10 7
4 �3 �2 1 �6 8
�2 8 3 �4 10 7
10 �5 �3 �18
6 � 4 �7 � 2 2 � (�5) �12 � (�6)
4 2 �5 �6
6 �7 2 �12
4 2�5 �6
6 �72 �12
a � j b � k c � l d � m e � n f � o g � p h � q i � r
j k l m n o p q r
a b c d e f g h i
a � j b � k c � l d � m e � n f � o g � p h � q i � r
j k l m n o p q r
a b c d e f g h i
Example 1Example 1
Example 2Example 2
ExercisesExercises
Scalar Multiplication You can multiply an m � n matrix by a scalar k.
Scalar Multiplication k �
If A � and B � , find 3B � 2A.
3B � 2A � 3 � 2 Substitution
� � Multiply.
� � Simplify.
� Subtract.
� Simplify.
Perform the indicated matrix operations. If the matrix does not exist, writeimpossible.
1. 6 2. � 3. 0.2
4. 3 � 2 5. �2 � 4
6. 2 � 5 7. 4 � 2
8. 8 � 3 9. � � � 3 �1 ��
32
� �74
�
3 �5 1 7
9 1 �7 0
1�4
28 8 18 1�7 20
4 0 �2 3 3 �4
2 1 3 �1 �2 4
�4 �14 28�16 26 6
4 3 �4 2 �5 �1
1 �2 5 �3 4 1
22 �1510 31
2 1 4 3
6 �10 �5 8
�14 2 8 6
�2 0 2 5
3 �1 0 7
�10 11 12 �1
�1 2 �3 5
�4 5 2 3
5 �2 �9 1 11 �612 7 �19
�2 �5 �3�17 11 �8 6 �1 �15
12 �30 18 0 42 �6�24 36 54
25 �10 �45 5 55 �30 60 35 �95
6 15 9 51 �33 24 �18 3 45
1�3
2 �5 3 0 7 �1 �4 6 9
�11 15 33 18
�3 � 8 15 � 0 21 � (�12) 24 � 6
8 0 �12 6
�3 15 21 24
2(4) 2(0) 2(�6) 2(3)
3(�1) 3(5) 3(7) 3(8)
4 0 �6 3
�1 5 7 8
�1 5 7 8
4 0�6 3
ka kb kc kd ke kf
a b c d e f
© Glencoe/McGraw-Hill 176 Glencoe Algebra 2
Study Guide and Intervention (continued)
Operations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
ExampleExample
ExercisesExercises
Skills PracticeOperations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
© Glencoe/McGraw-Hill 177 Glencoe Algebra 2
Less
on
4-2
Perform the indicated matrix operations. If the matrix does not exist, writeimpossible.
1. [5 �4] � [4 5] [9 1] 2. �
3. [3 1 6] � impossible 4. �
5. 3[9 4 �3] [27 12 �9] 6. [6 �3] � 4[4 7] [�10 �31]
7. �2 � 8. 3 � 4
9. 5 � 2 10. 3 � 2
Use A � , B � , and C � to find the following.
11. A � B 12. B � C
13. B � A 14. A � B � C
15. 3B 16. �5C
17. A � 4C 18. 2B � 3A 13 1014 5
15 �14�8 �1
15 �20�15 �5
6 63 �6
2 88 2
�1 0�3 �5
5 �2�2 �3
5 45 1
�3 4 3 1
2 21 �2
3 24 3
7 5 �1�24 9 21
1 �1 5 6 6 �3
3 1 3 �4 7 5
�8 40 44 1�3 5
6 5 �3 �2 1 0
�4 6 10 1 �1 1
16 �8�49
2 2 10
8 0 �3
5 �9�9 �17
1 1 1 1
�2 5 5 9
14 8 4 5 14 �2
9 9 2 4 6 4
5 �1 2 1 8 �6
4 �1 2
8 10�7 �3
0 �7 6 2
8 3 �1 �1
© Glencoe/McGraw-Hill 178 Glencoe Algebra 2
Perform the indicated matrix operations. If the matrix does not exist, writeimpossible.
1. � 2. �
3. �3 � 4 4. 7 � 2
5. �2 � 4 � 6. �
Use A � , B � , and C � to find the following.
7. A � B 8. A � C
9. �3B 10. 4B � A
11. �2B � 3C 12. A � 0.5C
ECONOMICS For Exercises 13 and 14, use the table that shows loans by an economicdevelopment board to women and men starting new businesses.
13. Write two matrices that represent the number of newbusinesses and loan amounts,one for women and one for men.
,
14. Find the sum of the numbers of new businesses and loan amounts for both men and women over the three-year period expressed as a matrix.
15. PET NUTRITION Use the table that gives nutritional information for two types of dog food. Find thedifference in the percent of protein, fat, and fiberbetween Mix B and Mix A expressed as a matrix.[2 �4 3]
% Protein % Fat % Fiber
Mix A 22 12 5
Mix B 24 8 8
63 1,431,00073 1,574,00063 1,339,000
36 864,00032 672,00028 562,000
27 567,00041 902,00035 777,000
Women Men
BusinessesLoan
BusinessesLoan
Amount ($) Amount ($)
1999 27 $567,000 36 $864,000
2000 41 $902,000 32 $672,000
2001 35 $777,000 28 $562,000
9 �5 3�6 4 12
�26 16 �28 16 12 �42
�12 17 20 7 �6 �38
6 �12 �15�3 0 27
�6 7 �6 3 10 �18
6 �5 �5�4 6 11
10 �8 6�6 �4 20
�2 4 5 1 0 �9
4 �1 0�3 6 2
24 324 3
27 �9 54 �18
2�3
8 12 �16 20
3�4
�12 �2
10 18
0 5
1 2
�1 4 �3 7 2 �6
2 �1 8 4 7 9
�9 64�135 81
�3 16 �21 12
�1 0 17 �11
71�116 42
�67 45 �24
4 �71 18
�4 8 10 �4 6 8
�6 9 7 �11 �8 17
2 �1 3 7 14 �9
Practice (Average)
Operations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
16 �15 6214 45 75
Reading to Learn MathematicsOperations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
© Glencoe/McGraw-Hill 179 Glencoe Algebra 2
Less
on
4-2
Pre-Activity How can matrices be used to calculate daily dietary needs?
Read the introduction to Lesson 4-2 at the top of page 160 in your textbook.
• Write a sum that represents the total number of Calories in the patient’sdiet for Day 2. (Do not actually calculate the sum.) 482 � 622 � 987
• Write the sum that represents the total fat content in the patient’s dietfor Day 3. (Do not actually calculate the sum.) 11 � 12 � 38
Reading the Lesson
1. For each pair of matrices, give the dimensions of the indicated sum, difference, or scalarproduct. If the indicated sum, difference, or scalar product does not exist, write impossible.
A � B �
C � D �
A � D: C � D: 5B:
�4C: 2D � 3A:
2. Suppose that M, N, and P are nonzero 2 � 4 matrices and k is a negative real number.Indicate whether each of the following statements is true or false.
a. M � (N � P) � M � (P � N) true b. M � N � N � M false
c. M � (N � P) � (M � N) � P false d. k(M � N) � kM � kN true
Helping You Remember
3. The mathematical term scalar may be unfamiliar, but its meaning is related to the wordscale as used in a scale of miles on a map. How can this usage of the word scale help youremember the meaning of scalar?
Sample answer: A scale of miles tells you how to multiply the distancesyou measure on a map by a certain number to get the actual distancebetween two locations. This multiplier is often called a scale factor. Ascalar represents the same idea: It is a real number by which a matrixcan be multiplied to change all the elements of the matrix by a uniformscale factor.
2 � 33 � 2
2 � 2impossible2 � 3
�3 6 0 �8 4 0
5 10 �3 6 4 12
�4 0 0 �5
3 5 6 �2 8 1
© Glencoe/McGraw-Hill 180 Glencoe Algebra 2
Sundaram’s SieveThe properties and patterns of prime numbers have fascinated many mathematicians.In 1934, a young East Indian student named Sundaram constructed the following matrix.
4 7 10 13 16 19 22 25 . . .7 12 17 22 27 32 37 42 . . .
10 17 24 31 38 45 52 59 . . .13 22 31 40 49 58 67 76 . . .16 27 38 49 60 71 82 93 . . .. . . . . . . . . . .
A surprising property of this matrix is that it can be used to determine whether or not some numbers are prime.
Complete these problems to discover this property.
1. The first row and the first column are created by using an arithmetic pattern. What is the common difference used in the pattern?
2. Find the next four numbers in the first row.
3. What are the common differences used to create the patterns in rows 2, 3,4, and 5?
4. Write the next two rows of the matrix. Include eight numbers in each row.
5. Choose any five numbers from the matrix. For each number n, that you chose from the matrix, find 2n � 1.
6. Write the factorization of each value of 2n � 1 that you found in problem 5.
7. Use your results from problems 5 and 6 to complete this statement: If n
occurs in the matrix, then 2n � 1 (is/is not) a prime number.
8. Choose any five numbers that are not in the matrix. Find 2n � 1 for each of these numbers. Show that each result is a prime number.
9. Complete this statement: If n does not occur in the matrix, then 2n � 1 is
.a prime number
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
Study Guide and InterventionMultiplying Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 181 Glencoe Algebra 2
Less
on
4-3
Multiply Matrices You can multiply two matrices if and only if the number of columnsin the first matrix is equal to the number of rows in the second matrix.
Multiplication of Matrices � �
Find AB if A � and B � .
AB � � Substitution
� Multiply columns by rows.
� Simplify.
Find each product, if possible.
1. � 2. � 3. �
4. � 5. � 6. �
7. � [0 4 �3] 8. � 9. �
not possible10 �1618 �6 5 9
11 �2113 �15
2 �2 3 1 �2 4
2 0 �3 1 4 �2 �1 3 1
1 �3 �2 0
7 �2 5 �4
6 10 �4 3 �2 7
24 �1514 �17
�1 4 8 4�3 �9
�15 1 7 26 �2 �12
4 �1 �2 5
5 �2 2 �3
1 2 2 1
3 �2 0 4 �5 1
4 0 �2 �3 1 1
�3 1 5 �2
7 �714 14
�3 �2 2 34
12 3�6 9
3 �1 2 4
3 �1 2 4
3 2 �1 4
�1 0 3 7
3 0 0 3
4 1 �2 3
�23 17 12 �10 �2 19
�4(5) � 3(�1) �4(�2) � 3(3) 2(5) � (�2)(�1) 2(�2) � (�2)(3) 1(5) � 7(�1) 1(�2) � 7(3)
5 �2 �1 3
�4 3 2 �2 1 7
5 �2�1 3
�4 3 2 �2 1 7
a1x1 � b1x2 a1y1 � b1y2 a2x1 � b2x2 a2y1 � b2y2
x1 y1 x2 y2
a1 b1 a2 b2
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 182 Glencoe Algebra 2
Multiplicative Properties The Commutative Property of Multiplication does not holdfor matrices.
Properties of Matrix MultiplicationFor any matrices A, B, and C for which the matrix product isdefined, and any scalar c, the following properties are true.
Associative Property of Matrix Multiplication (AB)C � A(BC)
Associative Property of Scalar Multiplication c(AB) � (cA)B � A(cB)
Left Distributive Property C(A � B) � CA � CB
Right Distributive Property (A � B)C � AC � BC
Use A � , B � , and C � to find each product.
a. (A � B)C
(A � B)C � � � � �
� �
�
�
b. AC � BC
AC � BC � � � �
� �
� � �
Note that although the results in the example illustrate the Right Distributive Property,they do not prove it.
Use A � , B � , C � , and scalar c � �4 to determine whether
each of the following equations is true for the given matrices.
1. c(AB) � (cA)B yes 2. AB � BA no
3. BC � CB no 4. (AB)C � A(BC) yes
5. C(A � B) � AC � BC no 6. c(A � B) � cA � cB yes
��12� �2
1 �3
6 42 1
3 25 �2
�12 �21 �5 �20
2 �4 �13 �19
�14 �17 8 �1
2(1) � 0(6) 2(�2) � 0(3) 5(1) � (�3)(6) 5(�2) � (�3)(3)
4(1) � (�3)(6) 4(�2) � (�3)(3) 2(1) � 1(6) 2(�2) � 1(3)
1 �2 6 3
2 0 5 �3
1 �2 6 3
4 �3 2 1
�12 �21 �5 �20
6(1) � (�3)(6) 6(�2) � (�3)(3) 7(1) � (�2)(6) 7(�2) � (�2)(3)
1 �2 6 3
6 �3 7 �2
1 �2 6 3
2 0 5 �3
4 �3 2 1
1 �26 3
2 05 �3
4 �32 1
Study Guide and Intervention (continued)
Multiplying Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
ExampleExample
ExercisesExercises
Skills PracticeMultiplying Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 183 Glencoe Algebra 2
Less
on
4-3
Determine whether each matrix product is defined. If so, state the dimensions ofthe product.
1. A2 � 5 � B5 � 1 2 � 1 2. M1 � 3 � N3 � 2 1 � 2
3. B3 � 2 � A3 � 2 undefined 4. R4 � 4 � S4 � 1 4 � 1
5. X3 � 3 � Y3 � 4 3 � 4 6. A6 � 4 � B4 � 5 6 � 5
Find each product, if possible.
7. [3 2] � [8] 8. �
9. � 10. � not possible
11. [�3 4] � [8 11] 12. � [2 �3 �2]
13. � not possible 14. �
15. � 16. �
Use A � , B � , C � , and scalar c � 2 to determine whether the following equations are true for the given matrices.
17. (AC)c � A(Cc) yes 18. AB � BA no
19. B(A � C) � AB � BC no 20. (A � B)c � Ac � Bc yes
3 �11 0
�3 2 5 1
2 12 1
44
2 2 2
0 1 1 1 1 0
�12 20 �6 8 6 0
3 �3 0 2
�4 4 �2 1 2 3
�6 6 15 12 3 �9
0 3 3 0
2 �2 4 5 �3 1
48
5 6 �3
�2 3 2 6 �9 �6
�1 3
0 �1 2 2
1 3 �1 1
3 �2
�3�5
3 �2
1 3 �1 1
28 �19 7 �9
2 �5 3 1
5 6 2 1
2 1
© Glencoe/McGraw-Hill 184 Glencoe Algebra 2
Determine whether each matrix product is defined. If so, state the dimensions ofthe product.
1. A7 � 4 � B4 � 3 7 � 3 2. A3 � 5 � M5 � 8 3 � 8 3. M2 � 1 � A1 � 6 2 � 6
4. M3 � 2 � A3 � 2 undefined 5. P1 � 9 � Q9 � 1 1 � 1 6. P9 � 1 � Q1 � 9 9 � 9
Find each product, if possible.
7. � 8. �
9. � 10. � not possible
11. [4 0 2] � [2] 12. � [4 0 2]
13. � 14. [�15 �9] � [�297 �75]
Use A � , B � , C � , and scalar c � 3 to determine whether the following equations are true for the given matrices.
15. AC � CA yes 16. A(B � C) � BA � CA no
17. (AB)c � c(AB) yes 18. (A � C)B � B(A � C) no
RENTALS For Exercises 19–21, use the following information.For their one-week vacation, the Montoyas can rent a 2-bedroom condominium for$1796, a 3-bedroom condominium for $2165, or a 4-bedroom condominium for$2538. The table shows the number of units in each of three complexes.
19. Write a matrix that represents the number of each type of unit available at each complex and a matrix that ,represents the weekly charge for each type of unit.
20. If all of the units in the three complexes are rented for the week at the rates given the Montoyas, express the income of each of the three complexes as a matrix.
21. What is the total income of all three complexes for the week? $526,054
172,452227,960125,642
$1796$2165$2538
36 24 2229 32 4218 22 18
2-Bedroom 3-Bedroom 4-Bedroom
Sun Haven 36 24 22
Surfside 29 32 42
Seabreeze 18 22 18
�1 0 0 �1
4 0�2 �1
1 33 1
6 11 23 �10
�30 10 15 �5
5 0 0 5
�6 2 3 �1
4 0 2 12 0 6�4 0 �2
1 3 �1
1 3 �1
3 �2 7 6 0 �5
3 �2 7 6 0 �5
�6 �12 39 3
2 4 7 �1
�3 0 2 5
2 20�23 �5
�3 0 2 5
2 4 7 �1
30 �4 �6 3 �6 26
3 �2 7 6 0 �5
2 4 3 �1
Practice (Average)
Multiplying Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
Reading to Learn MathematicsMultiplying Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 185 Glencoe Algebra 2
Less
on
4-3
Pre-Activity How can matrices be used in sports statistics?
Read the introduction to Lesson 4-3 at the top of page 167 in your textbook.
Write a sum that shows the total points scored by the Oakland Raidersduring the 2000 season. (The sum will include multiplications. Do notactually calculate this sum.)
6 � 58 � 1 � 56 � 3 � 23 � 2 � 1 � 2 � 2
Reading the Lesson
1. Determine whether each indicated matrix product is defined. If so, state the dimensionsof the product. If not, write undefined.
a. M3 � 2 and N2 � 3 MN: NM:
b. M1 � 2 and N1 � 2 MN: NM:
c. M4 � 1 and N1 � 4 MN: NM:
d. M3 � 4 and N4 � 4 MN: NM:
2. The regional sales manager for a chain of computer stores wants to compare the revenuefrom sales of one model of notebook computer and one model of printer for three storesin his area. The notebook computer sells for $1850 and the printer for $175. The numberof computers and printers sold at the three stores during September are shown in thefollowing table.
Write a matrix product that the manager could use to find the total revenue forcomputers and printers for each of the three stores. (Do not calculate the product.)
�
Helping You Remember
3. Many students find the procedure of matrix multiplication confusing at first because itis unfamiliar. Think of an easy way to use the letters R and C to remember how tomultiply matrices and what the dimensions of the product will be. Sample answer:Just remember RC for “row, column.” Multiply each row of the first matrixby each column of the second matrix. The dimensions of the product arethe number of rows of the first matrix and the number of columns of thesecond matrix.
1850 175
128 101205 166 97 73
Store Computers Printers
A 128 101
B 205 166
C 97 73
undefined3 � 4
1 � 14 � 4
undefinedundefined
2 � 23 � 3
© Glencoe/McGraw-Hill 186 Glencoe Algebra 2
Fourth-Order DeterminantsTo find the value of a 4 � 4 determinant, use a method called expansion by minors.
First write the expansion. Use the first row of the determinant.Remember that the signs of the terms alternate.
� � � � � � � � � �Then evaluate each 3 � 3 determinant. Use any row.
� � � � � � � � � �� 2(�16 � 10) � �3(24) � 5(�6)� �52 � �102
� � � � � � � � � �� 6(�16 � 10) � �4(�6) � 3(�12)� �156 � �12
Finally, evaluate the original 4 � 4 determinant.
� �Evaluate each determinant.
1.
� �2.
� �3.
� �1 4 3 0
�2 �3 6 45 1 1 24 2 5 �1
3 3 3 32 1 2 14 3 �1 52 5 0 1
1 2 3 14 3 �1 02 �5 4 41 �2 0 2
� 6(�52) � 3(�102) � 2(�156) � 7(�12) � �846
6 �3 2 70 4 3 50 2 1 �46 0 �2 0
0 26 0� 3
0 16 �2� �4
0 4 30 2 16 0 �2
4 52 �4� 6
0 4 50 2 �46 0 0
0 16 �2� 5
0 �46 0
� �30 3 50 1 �46 �2 0
4 52 �4� �(�2)
4 3 52 1 �40 �2 0
0 4 30 2 16 0 �2
� 70 4 50 2 �46 0 0
� 20 3 50 1 �46 �2 0
� (�3)4 3 52 1 �40 �2 0
� 6
6 �3 2 70 4 3 50 2 1 �46 0 �2 0
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
Study Guide and InterventionTransformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
© Glencoe/McGraw-Hill 187 Glencoe Algebra 2
Less
on
4-4
Translations and Dilations Matrices that represent coordinates of points on a planeare useful in describing transformations.
Translation a transformation that moves a figure from one location to another on the coordinate plane
You can use matrix addition and a translation matrix to find the coordinates of thetranslated figure.
Dilation a transformation in which a figure is enlarged or reduced
You can use scalar multiplication to perform dilations.
Find the coordinates of the vertices of the image of �ABC with vertices A(�5, 4), B(�1, 5), and C(�3, �1) if it is moved 6 units to the right and 4 unitsdown. Then graph �ABC and its image �A�B�C�.
Write the vertex matrix for �ABC.
Add the translation matrix to the vertex matrix of �ABC.
� �
The coordinates of the vertices of �A�B�C� are A�(1, 0), B�(5, 1), and C�(3, �5).
For Exercises 1 and 2 use the following information. Quadrilateral QUAD withvertices Q(�1, �3), U(0, 0), A(5, �1), and D(2, �5) is translated 3 units to the leftand 2 units up.
1. Write the translation matrix.
2. Find the coordinates of the vertices of Q�U�A�D�.Q�(�4, �1), U �, (�3, 2), A�(2, 1), D �(�1, �3)
For Exercises 3–5, use the following information. The vertices of �ABC are A(4, �2), B(2, 8), and C(8, 2). The triangle is dilated so that its perimeter is one-fourth the original perimeter.
3. Write the coordinates of the vertices of �ABC in a vertex matrix.
4. Find the coordinates of the vertices of image �A�B�C�.
A��1, � �, B �� , 2�, C ��2, �5. Graph the preimage and the image.
1�2
1�2
1�2
4 2 8�2 8 2
�
�
�x
y
O
�3 �3 �3 �3 2 2 2 2
1 5 3 0 1 �5
6 6 6 �4 �4 �4
�5 �1 �3 4 5 �1
6 6 6 �4 �4 �4
�5 �1 �3 4 5 �1 x
y
OA�
B�
C�
AB
C
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 188 Glencoe Algebra 2
Reflections and Rotations
ReflectionFor a reflection over the: x-axis y-axis line y � x
Matricesmultiply the vertex matrix on the left by:
Rotation
For a counterclockwise rotation about the90° 180° 270°
Matrices
origin of:
multiply the vertex matrix on the left by:
Find the coordinates of the vertices of the image of �ABC with A(3, 5), B(�2, 4), and C(1, �1) after a reflection over the line y � x.Write the ordered pairs as a vertex matrix. Then multiply the vertex matrix by thereflection matrix for y � x.
� �
The coordinates of the vertices of A�B�C� are A�(5, 3), B�(4, �2), and C�(�1, 1).
1. The coordinates of the vertices of quadrilateral ABCD are A(�2, 1), B(�1, 3), C(2, 2), andD(2, �1). What are the coordinates of the vertices of the image A�B�C�D� after areflection over the y-axis? A�(2, 1), B �(1, 3), C �(�2, 2), D �(�2, �1)
2. Triangle DEF with vertices D(�2, 5), E(1, 4), and F(0, �1) is rotated 90°counterclockwise about the origin.
a. Write the coordinates of the triangle in a vertex matrix.
b. Write the rotation matrix for this situation.
c. Find the coordinates of the vertices of �D�E�F�.D �(�5, �2), E �(�4, 1), F �(1, 0)
d. Graph �DEF and �D�E�F�.
�
�
� x
y
O
0 �11 0
�2 1 0 5 4 �1
5 4 �1 3 �2 1
3 �2 1 5 4 �1
0 1 1 0
0 1 �1 0
�1 0 0 �1
0 �1 1 0
0 1 1 0
�1 0 0 1
1 0 0 �1
Study Guide and Intervention (continued)
Transformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
ExampleExample
ExercisesExercises
Skills PracticeTransformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
© Glencoe/McGraw-Hill 189 Glencoe Algebra 2
Less
on
4-4
For Exercises 1–3, use the following information.Triangle ABC with vertices A(2, 3), B(0, 4), and C(�3, �3) is translated 3 units right and 1 unit down.
1. Write the translation matrix.
2. Find the coordinates of �A�B�C�. A�(5, 2), B�(3, 3), C �(0, �4)3. Graph the preimage and the image.
For Exercises 4–6, use the following information.The vertices of �RST are R(�3, 1), S(2, �1), and T(1, 3). The triangle is dilated so that its perimeter is twice the original perimeter.
4. Write the coordinates of �RST in a vertex matrix.
5. Find the coordinates of the image �R�S�T�.R �(�6, 2), S �(4, �2), T �(2, 6)
6. Graph �RST and �R�S�T�.
For Exercises 7–10, use the following information.The vertices of �DEF are D(4, 0), E(0, �1), and F(2, 3).The triangle is reflected over the x-axis.
7. Write the coordinates of �DEF in a vertex matrix.
8. Write the reflection matrix for this situation.
9. Find the coordinates of �D�E�F �. D �(4, 0), E �(0, 1), F �(2, �3)
10. Graph �DEF and �D�E�F �.
For Exercises 11–14, use the following information.Triangle XYZ with vertices X(1, �3), Y(�4, 1), and Z(�2, 5) is rotated 180º counterclockwise about the origin.
11. Write the coordinates of the triangle in a vertex matrix.
12. Write the rotation matrix for this situation.
13. Find the coordinates of �X�Y�Z�.X �(�1, 3), Y �(4, �1), Z�(2, �5)
14. Graph the preimage and the image.
�1 0 0 �1
1 �4 �2�3 1 5
�
�
�
x
y
O
1 00 �1 �
�
�
x
y
O
4 0 20 �1 3
�3 2 1 1 �1 3 �
�
�
x
y
O
3 3 3�1 �1 �1
�
�
�
x
y
O
© Glencoe/McGraw-Hill 190 Glencoe Algebra 2
For Exercises 1–3, use the following information.Quadrilateral WXYZ with vertices W(�3, 2), X(�2, 4), Y(4, 1), and Z(3, 0) is translated 1 unit left and 3 units down.
1. Write the translation matrix.
2. Find the coordinates of quadrilateral W�X�Y�Z�.W�(�4, �1), X �(�3, 1), Y �(3, �2), Z �(2, �3)
3. Graph the preimage and the image.
For Exercises 4–6, use the following information.The vertices of �RST are R(6, 2), S(3, �3), and T(�2, 5). The triangle is dilated so that its perimeter is one half the original perimeter.
4. Write the coordinates of �RST in a vertex matrix.
5. Find the coordinates of the image �R�S�T�.R �(3, 1), S �(1.5, �1.5), T �(�1, 2.5)
6. Graph �RST and �R�S�T�.
For Exercises 7–10, use the following information.The vertices of quadrilateral ABCD are A(�3, 2), B(0, 3), C(4, �4),and D(�2, �2). The quadrilateral is reflected over the y-axis.
7. Write the coordinates of ABCD in a vertex matrix.
8. Write the reflection matrix for this situation.
9. Find the coordinates of A�B�C�D�.A�(3, 2), B �(0, 3), C �(�4, �4), D �(2, �2)
10. Graph ABCD and A�B�C�D�.
11. ARCHITECTURE Using architectural design software, the Bradleys plot their kitchenplans on a grid with each unit representing 1 foot. They place the corners of an island at(2, 8), (8, 11), (3, 5), and (9, 8). If the Bradleys wish to move the island 1.5 feet to theright and 2 feet down, what will the new coordinates of its corners be?(3.5, 6), (9.5, 9), (4.5, 3), and (10.5, 6)
12. BUSINESS The design of a business logo calls for locating the vertices of a triangle at(1.5, 5), (4, 1), and (1, 0) on a grid. If design changes require rotating the triangle 90ºcounterclockwise, what will the new coordinates of the vertices be?(�5, 1.5), (�1, 4), and (0, 1)
�1 0 0 1
��
�
�
x
y
O
�3 0 4 �2 2 3 �4 �2
6 3 �22 �3 5
�
�
x
y
O
�1 �1 �1 �1�3 �3 �3 �3
�
�
�
�
x
y
O
Practice (Average)
Transformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
Reading to Learn MathematicsTransformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
© Glencoe/McGraw-Hill 191 Glencoe Algebra 2
Less
on
4-4
Pre-Activity How are transformations used in computer animation?
Read the introduction to Lesson 4-4 at the top of page 175 in your textbook.
Describe how you can change the orientation of a figure without changingits size or shape.
Flip (or reflect) the figure over a line.
Reading the Lesson
1. a. Write the vertex matrix for the quadrilateral ABCD shown in the graph at the right.
b. Write the vertex matrix that represents the position of the quadrilateral A�B�C�D� thatresults when quadrilateral ABCD is translated 3 units to the right and 2 units down.
2. Describe the transformation that corresponds to each of the following matrices.
a. b.
counterclockwise rotation translation 4 units down and about the origin of 180� 3 units to the right
c. d.
reflection over the y-axis reflection over the line y � x
Helping You Remember
3. Describe a way to remember which of the reflection matrices corresponds to reflectionover the x-axis.
Sample answer: The only elements used in the reflection matrices are 0,1, and �1. For such a 2 � 2 matrix M to have the property that
M � � , the elements in the top row must be 1 and 0 (in that
order), and elements in the bottom row must be 0 and �1 (in that order).
x�y
xy
0 1 1 0
�1 0 0 1
3 3 3 �4 �4 �4
�1 0 0 �1
�1 5 4 1�1 1 �6 �5
�4 2 1 �2 1 3 �4 �3 x
y
O
A
B
CD
© Glencoe/McGraw-Hill 192 Glencoe Algebra 2
Properties of DeterminantsThe following properties often help when evaluating determinants.
• If all the elements of a row (or column) are zero, the value of the determinant is zero.
� 0
• Multiplying all the elements of a row (or column) by a constant is equivalent to multiplying the value of the determinant by the constant.
3 �
• If two rows (or columns) have equal corresponding elements, the value of thedeterminant is zero.
� 0
• The value of a determinant is unchanged if any multiple of a row (or column) is added to corresponding elements of another row (or column).
�
(Row 2 is added to row 1.)
• If two rows (or columns) are interchanged, the sign of the determinant is changed.
�
• The value of the determinant is unchanged if row 1 is interchanged with column 1,and row 2 is interchanged with column 2. The result is called the transpose.
�
Exercises 1–6
Verify each property above by evaluating the given determinants and give another example of the property.
5 3 �7 4
5 �7 3 4
�3 8 5 5
4 5 �3 8
6 2 2 5
4 �3 2 5
5 5 �3 �3
12 �3 5 3
4 �1 5 3
a b0 0
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
Study Guide and InterventionDeterminants
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 193 Glencoe Algebra 2
Less
on
4-5
Determinants of 2 � 2 Matrices
Second-Order Determinant For the matrix , the determinant is � ad � bc.
Find the value of each determinant.
a.
� 6(5) � 3(�8)
� 30 �(�24) or 54
b.
� 11(�3) � (�5)(9)
� �33 � (�45) or 12
Find the value of each determinant.
1. 52 2. �2 3. �18
4. 64 5. �6 6. 1
7. �114 8. 144 9. �335
10. 48 11. �16 12. 495
13. 15 14. 36 15. 17
16. 18.3 17. 18. 37 6.8 15 �0.2 5
13�60
�23� ��
12�
�16� �
15�
4.8 2.1 3.4 5.3
20 110 0.1 1.4
8.6 0.5 14 5
0.2 8 �1.5 15
13 62 �4 19
24 �8 7 �3
10 16 22 40
�8 35 5 20
15 12 23 28
14 8 9 �3
4 7 5 9
�6 �3 �4 �1
5 12 �7 �4
3 9 4 6
�8 3 �2 1
6 �2 5 7
11 �5 9 3
11 �5 9 3
6 3 �8 5
6 3�8 5
a b c d
a b c d
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 194 Glencoe Algebra 2
Determinants of 3 � 3 Matrices
Third-Order Determinants � a � b � c
Area of a Triangle
The area of a triangle having vertices (a, b), (c, d ) and (e, f ) is |A |, where
A � .
Evaluate .
� 4 � 5 � 2 Third-order determinant
� 4(18 � 0) � 5(6 � 0) � 2(�3 � 6) Evaluate 2 � 2 determinants.
� 4(18) � 5(6) � 2(�9) Simplify.
� 72 � 30 � 18 Multiply.
� 60 Simplify.
Evaluate each determinant.
1. �57 2. 80 3. 28
4. 54 5. �63 6. �2
7. Find the area of a triangle with vertices X(2, �3), Y(7, 4), and Z(�5, 5).44.5 square units
5 �4 1 2 3 �2 �1 6 �3
6 1 �4 3 2 1 �2 2 �1
5 �2 2 3 0 �2 2 4 �3
6 1 4 �2 3 0 �1 3 2
4 1 0 �2 3 1 2 �2 5
3 �2 �2 0 4 1 �1 5 �3
1 3 2 �3
1 0 2 6
3 0 �3 6
4 5 �2 1 3 0 2 �3 6
4 5 �21 3 02 �3 6
a b 1 c d 1 e f 1
1�2
d e g h
d f g i
e f h i
a b c d e f g h i
Study Guide and Intervention (continued)
Determinants
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
ExampleExample
ExercisesExercises
Skills PracticeDeterminants
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 195 Glencoe Algebra 2
Less
on
4-5
Find the value of each determinant.
1. 13 2. 35 3. 1
4. �13 5. �45 6. 0
7. 14 8. 13 9. 1
10. 11 11. �5 12. �52
13. �3 14. 17 15. 68
16. �4 17. 10 18. �17
Evaluate each determinant using expansion by minors.
19. �1 20. �2 21. �1
Evaluate each determinant using diagonals.
22. �3 23. 8 24. 40 3 2 2 1 �1 4 3 �1 0
3 �1 2 1 0 4 3 �2 0
2 �1 6 3 2 5 2 3 1
2 6 1 3 5 �1 2 1 �2
6 �1 1 5 2 �1 1 3 �2
2 �1 1 3 2 �1 2 3 �2
�1 6 2 5
2 2 �1 4
�1 2 0 4
�1 �14 5 2
�1 �3 5 �2
3 �5 6 �11
�12 4 1 4
1 �3 �3 4
1 �5 1 6
9 �2 �4 1
�3 1 8 �7
�5 2 8 �6
3 12 2 8
0 9 5 8
2 5 3 1
1 6 1 7
10 9 5 8
5 2 1 3
© Glencoe/McGraw-Hill 196 Glencoe Algebra 2
Find the value of each determinant.
1. �5 2. 0 3. �18
4. 34 5. �20 6. 3
7. 36 8. 55 9. 112
10. 30 11. �16 12. 0.13
Evaluate each determinant using expansion by minors.
13. �48 14. 45 15. 7
16. 28 17. �72 18. 318
Evaluate each determinant using diagonals.
19. 10 20. 12 21. 5
22. 20 23. �4 24. 0
25. GEOMETRY Find the area of a triangle whose vertices have coordinates (3, 5), (6, �5),and (�4, 10). 27.5 units2
26. LAND MANAGEMENT A fish and wildlife management organization uses a GIS(geographic information system) to store and analyze data for the parcels of land itmanages. All of the parcels are mapped on a grid in which 1 unit represents 1 acre. Ifthe coordinates of the corners of a parcel are (�8, 10), (6, 17), and (2, �4), how manyacres is the parcel? 133 acres
2 1 3 1 8 0 0 5 �1
2 �1 �2 4 0 �2 0 3 2
1 2 �4 1 4 �6 2 3 3
1 �4 �1 1 �6 �2 2 3 1
2 2 3 1 �1 1 3 1 1
�4 3 �1 2 1 �2 4 1 �4
�12 0 3 7 5 �1 4 2 �6
2 7 �6 8 4 0 1 �1 3
0 �4 0 2 �1 1 3 �2 5
2 1 1 1 �1 �2 1 1 �1
2 �4 1 3 0 9 �1 5 7
�2 3 1 0 4 �3 2 5 �1
0.5 �0.7 0.4 �0.3
2 �13 �9.5
3 �4 3.75 5
�1 �11 10 �2
3 �4 7 9
4 0 �2 9
2 �5 5 �11
4 �3 �12 4
�14 �3 2 �2
4 1 �2 �5
9 6 3 2
1 6 2 7
Practice (Average)
Determinants
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
Reading to Learn MathematicsDeterminants
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 197 Glencoe Algebra 2
Less
on
4-5
Pre-Activity How are determinants used to find areas of polygons?
Read the introduction to Lesson 4-5 at the top of page 182 in your textbook.
In this lesson, you will learn how to find the area of a triangle if you know thecoordinates of its vertices using determinants. Describe a method you alreadyknow for finding the area of the Bermuda Triangle. Sample answer:Use the map. Choose any side of the triangle as the base, andmeasure this side with a ruler. Multiply this length by the scalefactor for the map. Next, draw a segment from the oppositevertex perpendicular to the base. Measure this segment, andmultiply its length by the scale for the map. Finally, find the area by using the formula A � bh.
Reading the Lesson
1. Indicate whether each of the following statements is true or false.
a. Every matrix has a determinant. false
b. If both rows of a 2 � 2 matrix are identical, the determinant of the matrix will be 0. true
c. Every element of a 3 � 3 matrix has a minor. true
d. In order to evaluate a third-order determinant by expansion by minors it is necessaryto find the minor of every element of the matrix. false
e. If you evaluate a third-order determinant by expansion about the second row, theposition signs you will use are � � �. true
2. Suppose that triangle RST has vertices R(�2, 5), S(4, 1), and T(0, 6).
a. Write a determinant that could be used in finding the area of triangle RST.
b. Explain how you would use the determinant you wrote in part a to find the area ofthe triangle. Sample answer: Evaluate the determinant and multiply the result by . Then take the absolute value to make sure the final answer
is positive.
Helping You Remember
3. A good way to remember a complicated procedure is to break it down into steps. Write alist of steps for evaluating a third-order determinant using expansion by minors.Sample answer: 1. Choose a row of the matrix. 2. Find the position signsfor the row you have chosen. 3. Find the minor of each element in thatrow. 4. Multiply each element by its position sign and by its minor. 5. Addthe results.
1�2
�2 5 1 4 1 1 0 6 1
1�2
© Glencoe/McGraw-Hill 198 Glencoe Algebra 2
Matrix MultiplicationA furniture manufacturer makes upholstered chairs and wood tables. Matrix Ashows the number of hours spent on each item by three different workers. One day the factory receives an order for 10 chairs and 3 tables. This is shown in matrix B.
� A � B
[10 3] � [10(4) � 3(18) 10(2) � 3(15) 10(12) � 3(0)] � [94 65 120]
The product of the two matrices shows the number of hours needed for each type of worker to complete the order: 94 hours for woodworking, 65 hours for finishing, and 120 hours for upholstering.
To find the total labor cost, multiply by a matrix that shows the hourly rate for each worker: $15 for woodworking, $9 for finishing, and $12 for upholstering.
C � [94 65 120] � [94(15) � 65(9) � 120(12)] � $3435
Use matrix multiplication to solve these problems.A candy company packages caramels, chocolates, and hard candy in three different assortments: traditional, deluxe, and superb. For each type of candy the table below gives the number in each assortment, the number of Calories per piece, and the cost to make each piece.
Calories cost per traditional deluxe superb per piece piece (cents)
caramels 10 16 15 60 10chocolates 12 8 25 70 12card candy 10 16 8 55 6
The company receives an order for 300 traditional, 180 deluxe and100 superb assortments.
1. Find the number of each type of candy needed to fill the order.7380 caramels; 7540 chocolates; 6680 hard candies
2. Find the total number of Calories in each type of assortment.1990-traditional; 2400-deluxe; 3090-superb
3. Find the cost of production for each type of assortment.$3.04-traditional; $3.52-deluxe; $4.98-superb
4. Find the cost to fill the order. $2043.60
15 9 12
4 2 12 18 15 0
chair tablenumber ordered [ 10 3 ]
hourswoodworker finsher upholsterer
chair 4 2 12 table 18 15 0
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
Study Guide and InterventionCramer’s Rule
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
© Glencoe/McGraw-Hill 199 Glencoe Algebra 2
Less
on
4-6
Systems of Two Linear Equations Determinants provide a way for solving systemsof equations.
Cramer’s Rule for
The solution of the linear system of equations ax � by � e
Two-Variable Systems
cx � dy � f
is (x, y ) where x � , y � , and � 0.
Use Cramer’s Rule to solve the system of equations. 5x � 10y � 810x � 25y � �2
x � Cramer’s Rule y �
� a � 5, b � �10, c � 10, d � 25, e � 8, f � �2 �
� Evaluate each determinant. �
� or Simplify. � � or �
The solution is � , � �.
Use Cramer’s Rule to solve each system of equations.
1. 3x � 2y � 7 2. x � 4y � 17 3. 2x � y � �22x � 7y � 38 (5, 4) 3x � y � 29 (9, �2) 4x � y � 4 (3, 8)
4. 2x � y � 1 5. 4x � 2y � 1 6. 6x � 3y � �35x � 2y � �29 (�3, �7) 5x � 4y � 24 �2, � � 2x � y � 21 (5, 11)
7. 2x � 7y � 16 8. 2x � 3y � �2 9. � � 2
x � 2y � 30 (22, �4) 3x � 4y � 9 (35, 24) � � �8 (�12, 30)
10. 6x � 9y � �1 11. 3x � 12y � �14 12. 8x � 2y �
3x � 18y � 12 9x � 6y � �75x � 4y � �
� , � �� , � �� , �11�14
1�7
5�6
4�3
5�9
2�3
27�7
3�7
y�6
x�4
y�5
x�3
7�2
2�5
4�5
2�5
90�225
4�5
180�225
5(�2) � 8(10)���5(25) � (�10)(10)
8(25) � (�2)(�10)���5(25) � (�10)(10)
5 8 10 �2�� 5 �10 10 25
8 �10 �2 25�� 5 �10 10 25
a e c f � a b c d
e b f d � a b c d
a b c d
a e c f � a b c d
e b f d � a b c d
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 200 Glencoe Algebra 2
Systems of Three Linear Equations
The solution of the system whose equations areax � by � cz � j
Cramer’s Rule for
dx � ey � fz � k
Three-Variable Systems
gx � hy � iz � l
is (x, y, z) where x � , y � , and z � and � 0.
Use Cramer’s rule to solve the system of equations.6x � 4y � z � 52x � 3y �2z � �28x � 2y � 2z � 10Use the coefficients and constants from the equations to form the determinants. Thenevaluate each determinant.
x � y � z �
� or � or � � or
The solution is � , � , �.
Use Cramer’s rule to solve each system of equations.
1. x � 2y � 3z � 6 2. 3x � y � 2z � �22x � y � z � �3 4x � 2y � 5z � 7x � y � z � 6 (1, 2, 3) x � y � z � 1 (3, �5, 3)
3. x � 3y � z � 1 4. 2x � y � 3z � �52x � 2y � z � �8 x � y � 5z � 214x � 7y � 2z � 11 (�2, 1, 6) 3x � 2y � 4z � 6 (4, 7, �2)
5. 3x � y � 4z � 7 6. 2x � y � 4z � 92x � y � 5z � �24 3x � 2y � 5z � �1310x � 3y � 2z � �2 (�3, 8, �2) x � y � 7z � 0 (5, 9, 2)
4�3
1�3
5�6
4�3
�128��96
1�3
32��96
5�6
�80��96
6 4 5 2 3 �2 8 �2 10�� 6 4 1 2 3 �2 8 �2 2
6 5 1 2 �2 �2 8 10 2�� 6 4 1 2 3 �2 8 �2 2
5 4 1 �2 3 �2 10 �2 2�� 6 4 1 2 3 �2 8 �2 2
a b c d e f g h i
a b j d e k g h l � a b c d e f g h i
a j c d k f g l i � a b c d e f g h i
j b c k e f l h i � a b c d e f g h i
Study Guide and Intervention (continued)
Cramer’s Rule
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
ExampleExample
ExercisesExercises
Skills PracticeCramer’s Rule
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
© Glencoe/McGraw-Hill 201 Glencoe Algebra 2
Less
on
4-6
Use Cramer’s Rule to solve each system of equations.
1. 2a � 3b � 6 2. 3x � y � 22a � b � �2 (�3, 4) 2x � y � 3 (1, �1)
3. 2m � 3n � �6 4. x � y � 2m � 3n � 6 (0, �2) 2x � 3y � 9 (3, 1)
5. 2x � y � 4 6. 3r � s � 77x � 2y � 3 (1, 2) 5r � 2s � 8 (6, 11)
7. 4g � 5h � 1 8. 7x � 5y � �8g � 3h � 2 (�1, 1) 9x � 2y � 3 (1, �3)
9. 3x � 4y � 2 10. 2x � y � 54x � 3y � 12 (6, 4) 3x � y � 5 (2, �1)
11. 3p � 6q � 18 12. x � 2y � �12p � 3q � 5 (4, �1) 2x � y � 3 (1, 1)
13. 5c � 3d � 5 14. 5t � 2v � 22c � 9d � 2 (1, 0) 2t � 3v � �8 (2, �4)
15. 5a � 2b � 14 16. 65w � 8z � 833a � 4b � 11 (3, 0.5) 9w � 4z � 0 (1, �2.25)
17. GEOMETRY The two sides of an angle are contained in the lines whose equations are3x � 2y � 4 and x � 3y � 5. Find the coordinates of the vertex of the angle. (2, �1)
Use Cramer’s Rule to solve each system of equations.
18. a � b � 5c � 2 19. x � 3y � z � 53a � b � 2c � 3 2x � 5y � z � 124a � 2b � c � �3 (2, �5, 1) x � 2y � 3z � �13 ( 3, 2, 4)
20. 3c � 5d � 2e � 4 21. r � 4s � t � 62c � 3d � 4c � �3 2r � s � 3t � 04c � 2d � 3e � 0 (1, �1, �2) 3r � 2s � t � 4 (1, �1, �1)
© Glencoe/McGraw-Hill 202 Glencoe Algebra 2
Use Cramer’s Rule to solve each system of equations.
1. 2x � y � 0 2. 5c � 9d � 19 3. 2x � 3y � 53x � 2y � �2 (2, �4) 2c � d � �20 (�7, 6) 3x � 2y � 1 (1, 1)
4. 20m � 3n � 28 5. x � 3y � 6 6. 5x � 6y � �452m � 3n � 16 (2, 4) 3x � y � �22 (�6, �4) 9x � 8y � 13 (�3, 5)
7. �2e � f � 4 8. 2x � y � �1 9. 8a � 3b � 24�3e � 5f � �15 (�5, �6) 2x � 4y � 8 (�2, �3) 2a � b � 4 (6, �8)
10. �3x � 15y � 45 11. 3u � 5v � 11 12. �6g � h � �10�2x � 7y � 18 (5, 4) 6u � 7v � �12 � , �2� �3g � 4h � 4 � , �2�
13. x � 3y � 8 14. 0.2x � 0.5y � �1 15. 0.3d � 0.6g � 1.8x � 0.5y � 3 (2, �2) 0.6x � 3y � �9 (5, 4) 0.2d � 0.3g � 0.5 (4, �1)
16. GEOMETRY The two sides of an angle are contained in the lines whose equations are
x � y � 6 and 2x � y � 1. Find the coordinates of the vertex of the angle. (2, �3)
17. GEOMETRY Two sides of a parallelogram are contained in the lines whose equationsare 0.2x � 0.5y � 1 and 0.02x � 0.3y � �0.9. Find the coordinates of a vertex of theparallelogram. (15, 4)
Use Cramer’s Rule to solve each system of equations.
18. x � 3y � 3z � 4 19. �5a � b � 4c � 7 20. 2x � y � 3z � �5�x � 2y � z � �1 �3a � 2b � c � 0 5x � 2y � 2z � 84x � y � 2z � �1 2a � 3b � c � 17 3x � 3y � 5z � 17
(1, �1, 2) (3, 2, �5) (2, 3, 4)
21. 2c � 3d � e � 17 22. 2j � k � 3m � �3 23. 3x � 2y � 5z � 34c � d � 5e � �9 3j � 2k � 4m � 5 2x � 2y � 4z � 3c � 2d � e � 12 �4j � k � 2m � 4 �5x � 10y � 7z � �3
(2, 3, �4) (�1, 2, 1) (1.2, 0.3, 0)
24. LANDSCAPING A memorial garden being planted in front of a municipal library will contain three circular beds that are tangent to each other. A landscape architect has prepared a sketch of the design for the garden using CAD (computer-aided drafting) software, as shown at theright. The centers of the three circular beds are representedby points A, B, and C. The distance from A to B is 15 feet,the distance from B to C is 13 feet, and the distance from A to C is 16 feet. What is the radius of each of the circularbeds? circle A: 9 ft, circle B: 6 ft, circle C: 7 ft
A
B C
13 ft
16 ft15 ft
4�3
4�3
1�3
Practice (Average)
Cramer’s Rule
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
Reading to Learn MathematicsCramer’s Rule
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
© Glencoe/McGraw-Hill 203 Glencoe Algebra 2
Less
on
4-6
Pre-Activity How is Cramer’s Rule used to solve systems of equations?
Read the introduction to Lesson 4-6 at the top of page 189 in your textbook.
A triangle is bounded by the x-axis, the line y � x, and the line
y � �2x � 10. Write three systems of equations that you could use to findthe three vertices of the triangle. (Do not actually find the vertices.)
x � 0, y � x; x � 0, y � �2x � 10; y � x, y � �2x � 10
Reading the Lesson
1. Suppose that you are asked to solve the following system of equations by Cramer’s Rule.
3x � 2y � 72x � 3y � 22
Without actually evaluating any determinants, indicate which of the following ratios ofdeterminants gives the correct value for x. B
A. B. C.
2. In your textbook, the statements of Cramer’s Rule for two variables and three variablesspecify that the determinant formed from the coefficients of the variables cannot be 0.If the determinant is zero, what do you know about the system and its solutions?
The system could be a dependent system and have infinitely manysolutions, or it could be an inconsistent system and have no solutions.
Helping You Remember
3. Some students have trouble remembering how to arrange the determinants that are usedin solving a system of two linear equations by Cramer’s Rule. What is a good way toremember this?
Sample answer: Let D be the determinant of the coefficients. Let Dx be the determinant formed by replacing the first column of D with the constantsfrom the right-hand side of the system, and let Dy be the determinantformed by replacing the second column of D with the constants. Then
x � and y � .Dy�D
Dx�D
3 7 2 22�� 3 2 2 �3
7 2 22 �3�� 3 2 2 �3
3 2 2 �3�� 7 2 22 �3
1�2
1�2
1�2
© Glencoe/McGraw-Hill 204 Glencoe Algebra 2
Communications NetworksThe diagram at the right represents a communications network linking five computer remote stations. The arrows indicate thedirection in which signals can be transmitted and received by eachcomputer. We can generate a matrix to describe this network.
A �
Matrix A is a communications network for direct communication. Suppose you want to send a message from one computer to another using exactly one other computer as a relay point. It can be shown that the entries of matrix A2 represent the number of ways to send a message from one point to another by going through a third station. For example, a message may be sent from station 1 to station 5 by going through station 2 or station 4 on the way. Therefore, A2
1,5 � 2.
A2 �
For each network, find the matrices A and A2. Then write the number of ways the messages can be sent for each matrix.
1. 2. 3.
A: A: A:
9 ways11 ways 10 ways
A: A: A:
21 ways21 ways 14 ways
1 0 0 0 00 1 0 0 01 1 1 0 01 1 0 1 03 1 1 1 0
1 0 1 0 1 00 1 0 0 1 11 0 2 1 0 00 1 1 2 0 01 1 0 0 2 10 1 0 1 0 0
3 1 2 01 2 1 11 1 2 11 2 1 1
0 1 0 0 01 0 0 0 01 0 0 1 01 0 1 0 01 1 1 1 0
0 1 0 1 0 00 0 1 0 0 00 1 0 0 1 11 0 0 0 1 00 0 1 1 0 01 0 0 0 0 0
0 1 1 11 0 1 01 1 0 01 0 1 0
2
5
1
3 4
2
6
1 3
4 5
1
2 3 4
Again, compare the entries of matrix A2 to thecommunications diagram to verify that the entries arecorrect. Matrix A2 represents using exactly one relay.
to computer j
1 1 1 1 2from
1 2 1 1 1computer i 1 2 1 1 1
1 2 0 4 1 1 1 1 1 2
The entry in position aij represents the number of ways to senda message from computer i to computer j directly. Compare theentries of matrix A to the diagram to verify the entries. Forexample, there is one way to send a message from computer 3to computer 4, so A3,4 � 1. A computer cannot send a messageto itself, so A1,1 � 0.
to computer j
0 1 0 1 0from
0 0 0 1 1computer i 1 0 0 1 0
1 1 1 0 1 0 1 0 1 0
1 2
3 4
5
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
Study Guide and InterventionIdentity and Inverse Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 205 Glencoe Algebra 2
Less
on
4-7
Identity and Inverse Matrices The identity matrix for matrix multiplication is asquare matrix with 1s for every element of the main diagonal and zeros elsewhere.
Identity Matrix If A is an n � n matrix and I is the identity matrix,for Multiplication then A � I � A and I � A � A.
If an n � n matrix A has an inverse A�1, then A � A�1 � A�1 � A � I.
Determine whether X � and Y � are inversematrices.Find X � Y.
X � Y � �
� or
Find Y � X.
Y � X � �
� or
Since X � Y � Y � X � I, X and Y are inverse matrices.
Determine whether each pair of matrices are inverses.
1. and 2. and 3. and
yes yes no
4. and 5. and 6. and
yes no yes
7. and 8. and 9. and
no yes no
10. and 11. and 12. and
no yes yes
�5 3 7 �4
4 3 7 5
5 �2 �17 7
7 2 17 5
3 2 �4 �3
3 2 4 �6
�72� ��
32�
1 �2 3 7 2 4
�3 4 2 ��
52�
5 8 4 6
��15� ��1
10�
�1
30� �1
10�
4 2 6 �2
�2 1 �
121� ��
52�
5 2 11 4
1 2 3 8
4 �1 5 3
�4 11 3 �8
8 11 3 4
2 3 �1 �2
2 3 5 �1
2 �1 ��
52� �
32�
3 2 5 4
4 �5 �3 4
4 5 3 4
1 0 0 1
21 � 20 12 � 12 �35 � 35 �20 � 21
7 4 10 6
3 �2 �5 �
72�
1 0 0 1
21 � 20 �14 � 14 30 � 30 �20 � 21
3 �2 �5 �
72�
7 4 10 6
3 �2 �5 �
72�
7 410 6
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 206 Glencoe Algebra 2
Find Inverse Matrices
Inverse of a 2 � 2 MatrixThe inverse of a matrix A � is
A�1 � , where ad � bc � 0.
If ad � bc � 0, the matrix does not have an inverse.
Find the inverse of N � .
First find the value of the determinant.
� 7 � 4 � 3
Since the determinant does not equal 0, N�1 exists.
N�1 � � �
Check:
NN�1 � � � �
N�1N � � � �
Find the inverse of each matrix, if it exists.
1. 2. 3.
no inverse exists
4. 5. 6.
��12
� no inverse exists 4 �2�10 �8
1�12
8 �5�10 6
8 2 10 4
3 6 4 8
6 5 10 8
30 1020 40
1�1000
1 �10 1
40 �10 �20 30
1 1 0 1
24 12 8 4
1 0 0 1
�73� � �
43� �
23� � �
23�
��
134� � �
134� ��
43� � �
73�
7 2 2 1
�13� ��
23�
��
23� �
73�
1 0 0 1
�73� � �
43� ��
134� � �
134�
�
23� � �
23� ��
43� � �
73�
�13� ��
23�
��
23� �
73�
7 2 2 1
�13� ��
23�
��
23� �
73�
1 �2 �2 7
1�3
d �b �c a
1�ad � bc
7 2 2 1
7 22 1
d �b �c a
1�ad � bc
a b c d
Study Guide and Intervention (continued)
Identity and Inverse Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
ExampleExample
ExercisesExercises
Skills PracticeIdentity and Inverse Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 207 Glencoe Algebra 2
Less
on
4-7
Determine whether each pair of matrices are inverses.
1. X � , Y � yes 2. P � , Q � yes
3. M � , N � no 4. A � , B � yes
5. V � , W � yes 6. X � , Y � yes
7. G � , H � yes 8. D � , E � no
Find the inverse of each matrix, if it exists.
9. � 10. �
11. no inverse exists 12.
13. 14. no inverse exists
15. 16. �
17. � 18. no inverse exists
19. � 20. 2 00 2
1�4
2 0 0 2
�8 �8�10 10
1�160
10 8 10 �8
10 8 5 4
0 77 0
1�49
0 �7 �7 0
2 �5�1 �4
1�13
�4 5 1 2
�1 1�1 �1
1�2
�1 �1 1 �1
3 6 �1 �2
3 1�3 1
1�6
1 �1 3 3
0 4�6 �2
1�24
�2 �4 6 0
9 3 6 2
2 �1�3 1
1 1 3 2
0 �2�4 0
1�8
0 2 4 0
�0.125 �0.125 �0.125 �0.125
�4 �4 �4 4
�121� �1
31�
��1
11� �1
41�
4 �3 1 2
��13� �
23�
�
16� �
16�
�1 4 1 2
0 ��17�
�
17� 0
0 7 �7 0
2 �5 1 �2
�2 5 �1 2
�1 0 0 �3
�1 0 0 3
�1 3 1 �2
2 3 1 1
1 0 �1 1
1 0 1 1
© Glencoe/McGraw-Hill 208 Glencoe Algebra 2
Determine whether each pair of matrices are inverses.
1. M � , N � no 2. X � , Y � yes
3. A � , B � yes 4. P � , Q � yes
Determine whether each statement is true or false.
5. All square matrices have multiplicative inverses. false
6. All square matrices have multiplicative identities. true
Find the inverse of each matrix, if it exists.
7. 8.
9. � 10.
11. 12. no inverse exists
GEOMETRY For Exercises 13–16, use the figure at the right.
13. Write the vertex matrix A for the rectangle.
14. Use matrix multiplication to find BA if B � .
15. Graph the vertices of the transformed triangle on the previous graph.Describe the transformation. dilation by a scale factor of 1.5
16. Make a conjecture about what transformation B�1 describes on a coordinate plane.
dilation by a scale factor of
17. CODES Use the alphabet table below and the inverse of coding matrix C � todecode this message:19 | 14 | 11 | 13 | 11 | 22 | 55 | 65 | 57 | 60 | 2 | 1 | 52 | 47 | 33 | 51 | 56 | 55.
CHECK_YOUR_ANSWERSCODE
A 1 B 2 C 3 D 4 E 5 F 6 G 7
H 8 I 9 J 10 K 11 L 12 M 13 N 14
O 15 P 16 Q 17 R 18 S 19 T 20 U 21
V 22 W 23 X 24 Y 25 Z 26 – 0
1 2 2 1
2�3
1.5 6 7.5 3 3 6 1.5 �1.5
1.5 0 0 1.5
1 4 5 22 4 1 �1
x
y
O
(2, –1)
(1, 2)
(4, 4)
(5, 1)
4 6 6 9 1 5
�3 21
�17
2 �5 3 1
3 �51 2
1�11
2 5 �1 3�7 �3
�4 �11�5
�1 3 4 �7
5 0�3 2
1�10
2 0 3 5�3 �5
4 41�8
4 5 �4 �3
�134� �
17�
�
17� �
37�
6 �2 �2 3
�15� ��1
10�
�
25� �1
30�
3 1 �4 2
3 2 5 3
�3 2 5 �3
�2 1 3 �2
2 1 3 2
Practice (Average)
Identity and Inverse Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
Reading to Learn MathematicsIdentity and Inverse Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 209 Glencoe Algebra 2
Less
on
4-7
Pre-Activity How are inverse matrices used in cryptography?
Read the introduction to Lesson 4-7 at the top of page 195 in your textbook.
Refer to the code table given in the introduction to this lesson. Suppose thatyou receive a message coded by this system as follows:
16 12 5 1 19 5 2 5 13 25 6 18 9 5 14 4.
Decode the message. Please be my friend.
Reading the Lesson
1. Indicate whether each of the following statements is true or false.
a. Every element of an identity matrix is 1. false
b. There is a 3 � 2 identity matrix. false
c. Two matrices are inverses of each other if their product is the identity matrix. true
d. If M is a matrix, M�1 represents the reciprocal of M. false
e. No 3 � 2 matrix has an inverse. true
f. Every square matrix has an inverse. false
g. If the two columns of a 2 � 2 matrix are identical, the matrix does not have aninverse. true
2. Explain how to find the inverse of a 2 � 2 matrix. Do not use any special mathematicalsymbols in your explanation.
Sample answer: First find the determinant of the matrix. If it is zero, thenthe matrix has no inverse. If the determinant is not zero, form a newmatrix as follows. Interchange the top left and bottom right elements.Change the signs but not the positions of the other two elements.Multiply the resulting matrix by the reciprocal of the determinant of theoriginal matrix. The resulting matrix is the inverse of the original matrix.
Helping You Remember
3. One way to remember something is to explain it to another person. Suppose that you arestudying with a classmate who is having trouble remembering how to find the inverse ofa 2 � 2 matrix. He remembers how to move elements and change signs in the matrix,but thinks that he should multiply by the determinant of the original matrix. How canyou help him remember that he must multiply by the reciprocal of this determinant?
Sample answer: If the determinant of the matrix is 0, its reciprocal isundefined. This agrees with the fact that if the determinant of a matrix is0, the matrix does not have an inverse.
© Glencoe/McGraw-Hill 210 Glencoe Algebra 2
Permutation MatricesA permutation matrix is a square matrix in which each row and each column has one entry that is 1.All the other entries are 0. Find the inverse of apermutation matrix interchanging the rows andcolumns. For example, row 1 is interchanged withcolumn 1, row 2 is interchanged with column 2.
P is a 4 � 4 permutation matrix. P�1 is the inverse of P.
Solve each problem.
1. There is just one 2 � 2 permutation 2. Find the inverse of the matrix you wrote matrix that is not also an identity in Exercise 1. What do you notice?matrix. Write this matrix.
The two matricesare the same.
3. Show that the two matrices in Exercises 1 and 2 are inverses.
4. Write the inverse of this matrix.
B � B 1
5. Use B�1 from problem 4. Verify that B and B�1 are inverses.
6. Permutation matrices can be used to write and decipher codes. To see how this is done, use the message matrix M and matrix B from problem 4. Find matrix C so that C equals the product MB. Use the rules below.
0 times a letter � 0
1 times a letter � the same letter
0 plus a letter � the same letter
7. Now find the product CB�1. What do you notice?
S H E S A W H I M
0 1 00 0 1
H E S A W S I M H
0 1 00 0 11 0 0
0 � 0 � 0 � 0 � 1 � 1 0 � 1 � 0 � 0 � 1 � 0 0 � 0 � 0 � 1 � 1 � 01 � 0 � 0 � 0 � 0 � 1 1 � 1 � 0 � 0 � 0 � 0 1 � 0 � 0 � 1 � 0 � 00 � 0 � 1 � 0 � 0 � 1 0 � 1 � 1 � 0 � 0 � 0 0 � 0 � 1 � 1 � 0 � 0
0 1 00 0 11 0 0
0 0 1 1 0 0 0 1 0
1 00 1
0 � 1 � 1 � 1 0 � 1 � 1 � 01 � 0 � 0 � 1 1 � 1 � 0 � 0
0 11 0
0 11 0
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
P � P�1 �
0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0
M � H E S A W S I M H
S H E S A W H I M
Study Guide and InterventionUsing Matrices to Solve Systems of Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
4-84-8
© Glencoe/McGraw-Hill 211 Glencoe Algebra 2
Less
on
4-8
Write Matrix Equations A matrix equation for a system of equations consists of theproduct of the coefficient and variable matrices on the left and the constant matrix on theright of the equals sign.
Write a matrix equation for each system of equations.
a. 3x � 7y � 12x � 5y � �8Determine the coefficient, variable, andconstant matrices.
� � 12 �8
x y
3 �7 1 5
b. 2x � y � 3z � �7x � 3y � 4z � 157x � 2y � z � �28
� � �7 15 �28
x y z
2 �1 3 1 3 �4 7 2 1
ExampleExample
ExercisesExercises
Write a matrix equation for each system of equations.
1. 2x � y � 8 2. 4x � 3y � 18 3. 7x � 2y � 155x � 3y � �12 x � 2y � 12 3x � y � �10
� � � � � �
4. 4x � 6y � 20 5. 5x � 2y � 18 6. 3x � y � 243x � y � 8� 0 x � �4y � 25 3y � 80 � 2x
� � � � � �
7. 2x � y � 7z � 12 8. 5x � y � 7z � 325x � y � 3z � 15 x � 3y � 2z � �18x � 2y � 6z � 25 2x � 4y � 3z � 12
� � � �
9. 4x � 3y � z � �100 10. x � 3y � 7z � 272x � y � 3z � �64 2x � y � 5z � 485x � 3y � 2z � 8 4x � 2y � 3z � 72
� � � �
11. 2x � 3y � 9z � �108 12. z � 45 � 3x � 2yx � 5z � 40 � 2y 2x � 3y � z � 603x � 5y � 89� 4z x � 4y � 2z � 120
� � � � 45 60120
xyz
3 �2 12 3 �11 �4 2
�108 40 89
xyz
2 3 �91 �2 53 5 �4
274872
xyz
1 �3 72 1 �54 �2 3
�100 �64 8
xyz
4 �3 �12 1 �35 3 �2
32�18 12
xyz
5 �1 71 3 �22 4 �3
121525
xyz
2 1 75 �1 31 2 �6
2480
xy
3 �12 3
1825
xy
5 21 4
20�8
xy
4 �63 1
15�10
xy
7 �23 1
1812
xy
4 �31 2
8�12
xy
2 15 �3
© Glencoe/McGraw-Hill 212 Glencoe Algebra 2
Solve Systems of Equations Use inverse matrices to solve systems of equationswritten as matrix equations.
Solving Matrix EquationsIf AX � B, then X � A�1B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Solve � .
In the matrix equation A � , X � , and B � .
Step 1 Find the inverse of the coefficient matrix.
A�1 � or .
Step 2 Multiply each side of the matrix equation by the inverse matrix.
� � � � Multiply each side by A�1.
� � Multiply matrices.
� Simplify.
The solution is (2, �2).
Solve each matrix equation or system of equations by using inverse matrices.
1. � � 2. � � 3. � �
(5, �3) no solution (13, �18)
4. � � 5. � � 6. � �
�� , � � (57, �31) �� , �7. 4x � 2y � 22 8. 2x � y � 2 9. 3x � 4y � 12
6x � 4y � �2 x � 2y � 46 5x � 8y � �8
(3, �5) (10, 18) (32, �21)
10. x � 3y � �5 11. 5x � 4y � 5 12. 3x � 2y � 52x � 7y � 8 9x � 8y � 0 x � 4y � 20
(�59, 18) � , � (�2, �5.5)45�76
10�19
15�7
9�7
3�2
1�4
3 �6
x y
1 2 3 �1
�15 6
x y
3 6 5 9
4 �8
x y
2 �3 2 5
3 �7
x y
3 2 5 4
16 12
x y
�4 �8 6 12
�2 18
x y
2 4 3 �1
2 �2
x y
16 �16
1�8
x y
1 0 0 1
6 4
4 �2 �6 5
1�8
x y
5 2 6 4
4 �2 �6 5
1�8
4 �2 �6 5
1�8
4 �2 �6 5
1�20 � 12
6 4
x y
5 2 6 4
64
x y
5 26 4
Study Guide and Intervention (continued)
Using Matrices to Solve Systems of Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
4-84-8
ExampleExample
ExercisesExercises
Skills PracticeUsing Matrices to Solve Systems of Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
4-84-8
© Glencoe/McGraw-Hill 213 Glencoe Algebra 2
Less
on
4-8
Write a matrix equation for each system of equations.
1. x � y � 5 2. 3a � 8b � 162x � y � 1 4a � 3b � 3
� � � �
3. m � 3n � �3 4. 2c � 3d � 64m � 3n � 6 3c � 4d � 7
� � � �
5. r � s � 1 6. x � y � 52r � 3s � 12 3x � 2y � 10
� � � �
7. 6x � y � 2z � �4 8. a � b � c � 5�3x � 2y � z � 10 3a � 2b � c � 0x � y � z � 3 2a � 3b � 8
� � � �
Solve each matrix equation or system of equations by using inverse matrices.
9. � � (2, �3) 10. � � (3, �2)
11. � � (3, �2) 12. � � (3, 2)
13. � � �7, � 14. � � �1, �
15. p � 3q � 6 16. �x � 3y � 22p � 3q � �6 (0, �2) �4x � 5y � 1 (1, �1)
17. 2m � 2n � �8 18. �3a � b � �96m � 4n � �18 (�1, �3) 5a � 2b � 14 (4, 3)
5�3
15 2
m n
5 6 12 �6
1�3
25 12
c d
3 12 2 �6
15 23
m n
7 �3 5 4
�1 7
a b
5 8 3 1
6 �3
x y
4 3 1 3
�7 �1
w z
1 3 4 3
508
abc
1 �1 13 2 �12 3 0
�4 10 3
xyz
6 �1 2�3 2 �1 1 1 1
510
xy
1 13 2
112
rs
1 �12 3
67
cd
2 33 �4
�3 6
m n
1 34 3
16 3
ab
3 84 3
51
xy
1 12 �1
© Glencoe/McGraw-Hill 214 Glencoe Algebra 2
Write a matrix equation for each system of equations.
1. �3x � 2y � 9 2. 6x � 2y � �25x � 3y � �13 3x � 3y � 10
� � � �
3. 2a � b � 0 4. r � 5s � 103a � 2b � �2 2r � 3s � 7
� � � �
5. 3x � 2y � 5z � 3 6. 2m � n � 3p � �5x � y � 4z � 2 5m � 2n � 2p � 8�2x � 2y � 7z � �5 3m � 3n � 5p � 17
� � � �
Solve each matrix equation or system of equations by using inverse matrices.
7. � � (2, �4) 8. � � (5, 1)
9. � � (3, �5) 10. � � (1, 7)
11. � � ��4, � 12. � � �� , �13. 2x � 3y � 5 14. 8d � 9f � 13
3x � 2y � 1 (1, 1) �6d � 5f � �45 (5, �3)
15. 5m � 9n � 19 16. �4j � 9k � �82m � n � �20 (�7, 6) 6j � 12k � �5 � , � �
17. AIRLINE TICKETS Last Monday at 7:30 A.M., an airline flew 89 passengers on acommuter flight from Boston to New York. Some of the passengers paid $120 for theirtickets and the rest paid $230 for their tickets. The total cost of all of the tickets was$14,200. How many passengers bought $120 tickets? How many bought $230 tickets?57; 32
18. NUTRITION A single dose of a dietary supplement contains 0.2 gram of calcium and 0.2 gram of vitamin C. A single dose of a second dietary supplement contains 0.1 gram of calcium and 0.4 gram of vitamin C. If a person wants to take 0.6 gram of calcium and1.2 grams of vitamin C, how many doses of each supplement should she take?2 doses of the first supplement and 2 doses of the second supplement
2�3
1�2
1�3
1�4
�1 �1
y z
8 3 12 6
1�2
17 �26
r s
�4 2 7 4
16 34
c d
�5 3 6 4
12 �11
a b
�1 �3 3 4
�7 10
x y
�2 3 1 5
0 �2
g h
2 1 3 2
�5 8 17
m n p
2 1 �35 2 �23 �3 5
3 2�5
xyz
3 �2 5 1 1 �4�2 2 7
10 7
rs
1 52 �3
0�2
ab
2 13 2
�2 10
xy
6 �23 3
9�13
xy
�3 2 5 �3
Practice (Average)
Using Matrices to Solve Systems of Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
4-84-8
Reading to Learn MathematicsUsing Matrices to Solve Systems of Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
4-84-8
© Glencoe/McGraw-Hill 215 Glencoe Algebra 2
Less
on
4-8
Pre-Activity How are inverse matrices used in population ecology?
Read the introduction to Lesson 4-8 at the top of page 202 in your textbook.
Write a 2 � 2 matrix that summarizes the information given in theintroduction about the food and territory requirements for the two species.
Reading the Lesson
1. a. Write a matrix equation for the following system of equations.3x � 5y � 10
� � 2x � 4y � �7
b. Explain how to use the matrix equation you wrote above to solve the system. Use asfew mathematical symbols in your explanation as you can. Do not actually solve thesystem.
Sample answer: Find the inverse of the 2 � 2 matrix of coefficients.Multiply this inverse by the 2 � 1 matrix of constants, with the 2 � 2matrix on the left. The product will be a 2 � 1 matrix. The number inthe first row will be the value of x, and the number in the second rowwill be the value of y.
2. Write a system of equations that corresponds to the following matrix equation.
� �3x � 2y � 4z � �22x � y � 65y � 6z � �4
Helping You Remember
3. Some students have trouble remembering how to set up a matrix equation to solve asystem of linear equations. What is an easy way to remember the order in which to writethe three matrices that make up the equation?
Sample answer: Just remember “CVC” for “coefficients, variables,constants.”The variable matrix is on the left side of the equals sign, justas the variables are in the system of linear equations. The constantmatrix is on the right side of the equals sign, just as the constants are inthe system of linear equations.
�2 6 �4
x y z
3 2 �4 2 �1 0 0 5 6
10�7
xy
3 52 �4
140 500120 400
© Glencoe/McGraw-Hill 216 Glencoe Algebra 2
Properties of MatricesComputing with matrices is different from computing with real numbers. Stated below are some properties of the real number system. Are these also true for matrices? In the problems on this page, you will investigate this question.
For all real numbers a and b, ab � 0 if and only if a � 0 or b � 0.
Multiplication is commutative. For all real numbers a and b, ab � ba.
Multiplication is associative. For all real numbers a, b, and c, a(bc) � (ab)c.
Use the matrices A, B, and C for the problems. Write whether each statement is true. Assume that a 2-by-2 matrix is the 0 matrix if and only if all of its elements are zero.
A � B � C �
1. AB � 0 2. AC � 0 3. BC � 0
4. AB � BA 5. AC � CA 6. BC � CB
7. A(BC) � (AB)C 8. B(CA) � (BC)A 9. B(AC) � (BA)C
10. Write a statement summarizing your findings about the properties of matrixmultiplication.Based on these examples, matrix multiplication is associative, but notcommutative. Two matrices may have a product of zero even if neither ofthe factors equals zero.
�8 �
3 6 1 2
1 �3 �1 3
3 1 1 3
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-84-8
Chapter 4 Test, Form 1
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 217 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. How many elements are there in a 3 � 4 matrix?A. 7 B. 3 C. 12 D. 4 1.
2. Solve the matrix equation � � � � � for y.
A. 1 B. 6 C. 7 D. �1 2.
For Questions 3–10, use the matrices to find the following.
P � � � Q � � � R � � � S � � �T � � � U � � � V � � �
3. the first row of T � UA. [�4 1 12] B. [1 �8 �9] C. [16 �9 6] D. not possible 3.
4. the first row of U � VA. [10 �3 �8] B. [�10 2] C. [10 �7 2] D. not possible 4.
5. the first row of 4TA. [�2 8 �5] B. [12 �4 �20] C. [24 �16 36] D. not possible 5.
6. the first row of 2P � SA. [9 �1] B. [10 �4] C. [9 5] D. not possible 6.
7. the first row of TVA. [12 �4 �20] B. [�23 21] C. [53 �27] D. not possible 7.
8. the inverse of matrix RA. P B. Q C. S D. not possible 8.
9. the dimensions of PQA. 1 � 2 B. 2 � 2 C. 2 � 1 D. 4 � 4 9.
10. the pair of matrices that are inversesA. P and S B. Q and S C. P and Q D. Q and R 10.
11. Find the value of � �.A. 13 B. 7 C. 17 D. 3 11.
12. Which expression is true for all matrices X2�2, Y2�2 and scalars c?A. c(X � Y) � (Y � X)c B. XY � YXC. c(XY) � (YX)c D. c(XY) � (cX)(cY) 12.
5 13 2
0 4�2 6
5 �3
10 �5 �32 7 4
6 �4 93 �1 �5
1 �30 �
12�
0 �12�
1 �21 60 2
4 12 0
3 61 z
3 xy 7
44
© Glencoe/McGraw-Hill 218 Glencoe Algebra 2
Chapter 4 Test, Form 1 (continued)
13. Evaluate � � using expansion by minors.
A. 5 B. �7 C. 7 D. �3 13.
14. Evaluate � � using diagonals.
A. �2 B. 7 C. 11 D. �1 14.
15. Triangle RST with vertices R(2, 5), S(1, 4), and T(3, 1) is translated 3 units right. What are the coordinates of S�?A. (4, 4) B. (4, 7) C. (1, 7) D. (�2, 4) 15.
16. The vertices of XYZ are X(3, 1), Y(0, 4), and Z(0, 0). The triangle is being
reflected over the line y � x. Use the reflection matrix � � to find X�.
A. (3, 1) B. (1, 3) C. (�3, 1) D. (3, �1) 16.
17. Cramer’s Rule is used to solve the system of equations 2m � 3n � 11 and 3m � 5n � 6. Which determinant represents the numerator for m?
A. � � B. � � C. � � D. � � 17.
18. Cramer’s Rule is used to solve the system of equations 2x � 3y � 4z � 12,3x � y � 5z � 10 and x � 4y � z � 8. Which determinant represents the numerator for y?
A. � � B. � � C. � � D. � � 18.
19. Which matrix would not be used to write a matrix equation for the system of equations 3f � 2g � 7 and �2f � g � �5?
A. � � B. � � C. � � D. � � 19.
20. Which product would be used to solve the matrix equation � � � � � � � �by using inverse matrices?
A. � � � � � B. �14�� � � � � C. �
14�� � � � � D. 4� � � � � 20.
Bonus Find the value of � �. B:0 1 0a b cc a b
40
1 �60 4
40
4 60 1
40
1 �60 4
40
4 60 1
40
mn
4 60 1
7�5
gf
3 �2�2 1
fg
2 4 123 5 101 �1 8
2 �3 43 1 51 �4 �1
12 �3 410 1 58 �4 �1
2 12 43 10 51 8 �1
11 36 �5
2 113 6
2 33 �5
11 26 3
0 11 0
2 0 13 1 21 �2 5
1 3 20 �1 12 4 1
NAME DATE PERIOD
44
Chapter 4 Test, Form 2A
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 219 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Solve the matrix equation � � � � � for x.
A. 20 B. 25 C. 1 D. 4 1.
For Questions 2–10, use the matrices to find the following.
P � � � Q � � � R � � � S � � �T � � � U � � � V � � �
2. the dimensions of matrix TA. 6 � 1 B. 3 � 2 C. 2 � 3 D. 1 � 6 2.
3. the first row of T � VA. [�9 6 8] B. [�9 �2] C. [�7 7 4] D. not possible 3.
4. the first row of U � TA. [14 �7 �7] B. [0] C. [10 �6 �13] D. not possible 4.
5. the first row of �4UA. [�4] B. [�16 36 20] C. [�12 0 8] D. not possible 5.
6. the first row of 4Q � PA. [4 �2] B. [4 �6] C. [4 22] D. not possible 6.
7. the first row of UVA. [2 21] B. [22 �36 �24] C. [0 14] D. not possible 7.
8. the inverse of matrix RA. P B. Q C. S D. not possible 8.
9. the dimensions of VTA. 2 � 3 B. 3 � 2 C. 3 � 3 D. 2 � 2 9.
10. the pair of matrices that are inversesA. P and Q B. P and S C. Q and S D. P and R 10.
11. Find the value of � �.A. 8 B. �64 C. 6 D. �8 11.
12. GEOMETRY Find the area of a triangle whose vertices have coordinates (�4, 3), (2, 5), and (�7, �1).A. 9 units2 B. 18 units2 C. 14 units2 D. 7 units2 12.
12 �47 �3
2 4�1 0
3 �1
3 0 �24 �9 �5
�11 7 5�6 �3 8
��34� �
12�
�114� 0
1 �12�
0 ��14�
1 20 �4
0 142 21
10 � 2y5 � x
3xy
44
© Glencoe/McGraw-Hill 220 Glencoe Algebra 2
Chapter 4 Test, Form 2A (continued)
13. Evaluate � � using diagonals.
A. 58 B. �2 C. 12 D. 10 13.
14. For all matrices X3�3, Y2�3, and Z3�3 and scalars q, which statement is always true?A. X � 2Z � 2X � Z B. q(XZ) � (qX)ZC. q(YZ) � (ZY)q D. (XY)Z � Z(YX) 14.
15. MAP On a map, the coordinates of the corners of a town are A(0.5, 2),B(2, 3.5), C(5, 1.5), and D(3, 1). The map is dilated so that the perimeter of the town is five times its original perimeter. Find the coordinates of C�.A. (25, 7.5) B. (1, 0.3) C. (25, 1.5) D. (10, 6.5) 15.
16. Triangle RST with vertices R(�10, �8), S(1, 7), and T(5, �10) is rotated 90�counterclockwise about the origin. Find the coordinates of T�.A. (�5, 10) B. (�10, �5) C. (10, 5) D. (10, �5) 16.
17. Cramer’s Rule is used to solve the system of equations 3m � 5n � 12 and 4m � 7n � �5. Which determinant represents the numerator for n?
A. � � B. � � C. � � D. � � 17.
18. Cramer’s Rule is used to solve the system of equations 3x � y � 2z � 17,4x � 2y � 3z � 10, and 2x � 5y � 9z � �6. Which determinant represents the numerator for z?
A. � � B. � � C. � � D. � � 18.
19. Which matrix would not be used to write a matrix equation for the system of equations 5m � 2n � 13 and �m � n � �2?
A. � � B. � � C. � � D. � � 19.
20. Which product would be used to solve the matrix equation
� � � � � � � � by using inverse matrices?
A. �111�� � � � � B. �1
11�� � � � � C. � � � � � D. � � � � � 20.
Bonus Find the value of � �. B:d f d1 1 1f d f
67
3 �42 1
67
1 4�2 3
67
3 �42 1
67
1 4�2 3
67
ab
3 �42 1
5 �2�1 1
5 13�1 �2
13�2
mn
3 17 24 10 �32 �6 �9
17 3 �110 4 2
�6 2 5
3 �1 174 2 102 5 �6
3 �1 24 2 �32 5 �9
12 �5�5 7
3 124 �5
3 �54 7
12 3�5 4
1 4 �10 3 52 6 �2
NAME DATE PERIOD
44
Chapter 4 Test, Form 2B
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 221 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Solve the matrix equation � � � � � for x.
A. 1 B. �1 C. 7 D. �7 1.
For Questions 2–10, use the matrices to find the following.
P � � � Q � � � R � � � S � � �T � � � U � � � V � � �
2. the dimensions of matrix VA. 6 � 1 B. 1 � 6 C. 2 � 3 D. 3 � 2 2.
3. the first row of T � UA. [�5 1 6] B. [13 �11 �2] C. [3 �3 6] D. not possible 3.
4. the first row of V � TA. [�1 �7] B. [�7 3 2] C. [�1 5 �6] D. not possible 4.
5. the first row of �3TA. [�12 �24] B. [�12 15 �6] C. [�24 3 �9] D. not possible 5.
6. the first row of 5P � QA. [11 1] B. [9 1] C. [19 9] D. not possible 6.
7. the first row of TVA. [20 �16 9] B. [20 24] C. [4 4] D. not possible 7.
8. the inverse of matrix RA. P B. Q C. S D. not possible 8.
9. the dimensions of STA. 1 � 6 B. 2 � 2 C. 3 � 2 D. 2 � 3 9.
10. the pair of matrices that are inversesA. S and R B. Q and R C. P and Q D. Q and S 10.
11. Find the value of � �.A. 6 B. �6 C. �54 D. 54 11.
12. GEOMETRY Find the area of a triangle whose vertices have coordinates (2, �5), (6, 1), and (�3, �4).A. 34 units2 B. 21 units2 C. 17 units2 D. 42 units2 12.
15 4�6 �2
3 10 2
�4 5
�9 6 4�5 �2 3
4 �5 28 �1 3
�172� ��
19�
��13� �
19�
0 ��14�
1 �34�
4 412 21
3 1�4 0
2x3y
44
© Glencoe/McGraw-Hill 222 Glencoe Algebra 2
Chapter 4 Test, Form 2B (continued)
13. Evaluate � � using diagonals.
A. –38 B. 94 C. �42 D. 114 13.
14. For all matrices A2�2, B2�4, and C2�2, and scalars d, which statement is always true?A. A � dC � dA � C B. (CB)d � d(BC)C. d(A � C) � dA � dC D. (A � C)B � B(A � C) 14.
15. MAP On a town map, the coordinates of the corners of the zoo are L(1.2, 4),M(2, 0.8), N(4, 1.6), and P(6, 6). The map is dilated so that the perimeter of the zoo is four times its original size. Find the coordinates of M�.A. (8, 0.8) B. (0.5, 0.1) C. (8, 3.2) D. (6, 4.8) 15.
16. Triangle EFG with vertices E(�4, �5), F(1, 3), and G(4, �1) is rotated 270�counterclockwise about the origin. Find the coordinates of G�.A. (1, 4) B. (�4, 1) C. (�1, 4) D. (�1, �4) 16.
17. Cramer’s Rule is used to solve the system of equations 5f � 9g � 10 and 4f � 3g � �6. Which determinant represents the numerator for f?
A. � � B. � � C. � � D. � � 17.
18. Cramer’s Rule is used to solve the system of equations 4x � 5y � z � 11,3x � 2y � 2z � �5, and 2x � 6y � 3z � 8. Which determinant represents the numerator for z?
A. � � B. � � C. � � D. � � 18.
19. Which matrix would not be used to write a matrix equation for the system of equations 2c � 5d � �11 and �c � 2d � 10?
A. �19�� � B. � � C. � � D. � � 19.
20. Which product would be used to solve the matrix equation
� � � � � � � � by using inverse matrices?
A. � � � � � B. �110�� � � � � C. �1
10�� � � � � D. � � � � � 20.
Bonus Find the value of � �. B:�a b �c
a �b ca 1 1
26
7 �31 1
26
7 �31 1
26
1 3�1 7
26
1 3�1 7
26
mn
7 �31 1
�1110
2 5�1 2
cd
2 �51 2
4 �5 13 �2 22 6 3
4 �5 113 �2 �52 6 8
4 11 �53 �5 �22 8 6
11 4 �5�5 3 �2
8 2 6
�9 103 �6
5 104 �6
5 �94 3
10 �9�6 3
2 �3 14 0 �25 �1 6
NAME DATE PERIOD
44
Chapter 4 Test, Form 2C
© Glencoe/McGraw-Hill 223 Glencoe Algebra 2
1. GOLF At a municipal golf course, the fees to play a round 1.of golf are as shown. Write a 3 � 2 matrix to organize this information.
2. Solve the matrix equation � � � � �. 2.
For Questions 3–6, perform the indicated matrix operations. If the matrix does not exist, write impossible.
3. � � � � � 3.
4. � � � � � 4.
5. 4[2 1 9 12] � 2[�5 8 7 0] 5.
6. 7� � � 5� � � 3� � 6.
7. Determine whether the matrix X3�5 � Y5�9 is defined. 7.If so, state the dimensions of the product.
8. Find � � � � �, if possible. 8.
9. Use A � � �, B � � �, and C � � � to find 9.
(AB)C and A(BC). Then state whether (AB)C � A(BC) is true for the given matrices.
10. Triangle ABC with vertices A(�4, �3), B(2, �5), and 10.C(�1, 4) is translated 3 units right and 2 units down. Find the coordinates of �A�B�C�.
0 �27 3
5 �3�6 1
3 10 �4
9 �8 12 �4 �6
4 �12 �5
104
�1
61
�2
30
�4
3 12 �117 2 8
1 �4 9 03 2 �5 6
10 �5 3 19 �4 0 �2
6 3 10 41 �7 �8 3
18�11
2x � 3y3x � 5y
NAME DATE PERIOD
SCORE 44
Ass
essm
ent
Resident Non-resident
Weekend
Weekday morning $20 $25
$40
Weekday afternoon $18 $22
$40
© Glencoe/McGraw-Hill 224 Glencoe Algebra 2
Chapter 4 Test, Form 2C (continued)
11. The vertices of parallelogram RSTU are R(�4, �2), S(0, �2), 11.T(6, 1), and U(2, 1). The parallelogram is reflected over the y-axis. Find the coordinates of parallelogram R�S�T�U�.
12. Find the value of � �. 12.
13. Evaluate � � using expansion by minors. 13.
14. GEOMETRY Find the area of a triangle whose vertices 14.are located at (�7, �1), (1, 6), and (5, �3).
For Questions 15 and 16, use Cramer’s Rule to solve each system of equations.
15. 3a � 2b � 6.5 15.2a � 1.5b � 10
16. 2x � 5y � 3z � 27 16.4x � 3y � 7z � �37x � 2y � 5z � 30
17. Determine whether P � � � and Q � � � are 17.
inverses.
18. Find the inverse of R � � �, if it exists. 18.
19. Solve the matrix equation � � � � � � � � by using 19.
inverse matrices.
20. TRANSPORTATION A transportation company rents cars, 20.trucks, and vans. The company has 366 vehicles in its fleet.It has twice as many trucks as vans, and 186 more cars than vans. Let c represent the number of cars, t represent the number of trucks, and v represent the number of vans.Write a matrix equation that describes this situation.
Bonus Find the value of � �. B:x �y 1
�x �y 11 0 1
54
xy
�2 4�1 3
4 22 4
��16� �
14�
�12� ��
14�
3 36 2
6 1 4�5 9 �3
2 �8 4
�7 124 �11
NAME DATE PERIOD
44
Chapter 4 Test, Form 2D
© Glencoe/McGraw-Hill 225 Glencoe Algebra 2
1. TICKETS At a local ballpark, ticket prices for home games 1.are as shown. Write a 3 � 2 matrix to organize this information.
2. Solve the matrix equation � � � � �. 2.
For Questions 3–6, perform the indicated matrix operations. If the matrix does not exist, write impossible.
3. � � � � � 3.
4. � � � � � 4.
5. 9[�1 3 7 2] � 6[4 5 �8 1] 5.
6. 4� � � 3� � � 6� � 6.
7. Determine whether the matrix P2�4 � Q4�7 is defined. 7.If so, state the dimensions of the product.
8. Find � � � � �, if possible. 8.
9. Use A � � �, B � � �, and scalar d � �2 to find 9.
d(AB) and (dA)B. Then state whether d(AB) � (dA)B is true for the given matrices.
10. Triangle ABC with vertices A(4, �7), B(2, 3), and C(�3, �1) 10.is translated 2 units left and 4 units up. Find the coordinates of �A�B�C�.
11 �72 1
4 9�5 2
3 �4 �98 7 10
2 1�3 �6
�137
92
�1
0�5
3
12 �6 98 4 5
�7 �3 9
4 �51 69 �2
�6 3 0 114 �7 12 5
9 �6 10 17 4 �8 0
�3412
3x � 7y4x � 5y
NAME DATE PERIOD
SCORE 44
Ass
essm
ent
Bleachers Box Seats
Double-Header
Weekday $9 $21
$27
Weekend $12 $26
$27
© Glencoe/McGraw-Hill 226 Glencoe Algebra 2
Chapter 4 Test, Form 2D (continued)
11. The vertices of parallelogram RSTU are R(�3, �5), S(4, �5), 11.T(6, 4), and U(�1, 4). The parallelogram is reflected over the x-axis. Find the coordinates of parallelogram R�S�T�U�.
12. Find the value of � �. 12.
13. Evaluate � � using expansion by minors. 13.
14. GEOMETRY Find the area of a triangle whose vertices are 14.located at (�8, 4), (2, 6), and (3, �2).
For Questions 15 and 16, use Cramer’s Rule to solve each system of equations.
15. 0.5x � 2y � 7 15.3x � 10y � 9
16. 4a � 3b � 5c � 2 16.2a � 4b � 7c � 20a � 3b � 8c � �13
17. Determine whether S � � � and T � � � are inverses. 17.
18. Find the inverse of A � � �, if it exists. 18.
19. Solve the matrix equation � � � � � � � � by using 19.
inverse matrices.
20. GARDENING A garden shop sells flowers, trees, and shrubs. 20.The shop sold 1085 items last month. Fifteen more shrubs were sold than trees, and 8 times as many flowers were sold as shrubs. Let f represent the number of flowers, t represent the number of trees, and s represent the number of shrubs.Write a matrix equation that describes this situation.
Bonus Find the value of � �. B:a b �c1 1 0a �b c
�72
xy
6 �8�1 3
4 �23 �1
�112� �
13�
�16� �
13�
4 42 �1
9 �5 81 2 14 �2 �2
10 �76 �5
NAME DATE PERIOD
44
Chapter 4 Test, Form 3
© Glencoe/McGraw-Hill 227 Glencoe Algebra 2
1. TICKETS A theater group is to put on four performances 1.of its latest production. Each ticket for a Friday or Saturday evening show will cost $10. Each ticket for a Saturday or Sunday matinee will cost $7. Write a 3 � 2 matrix to organize this information.
2. Solve the matrix equation � � � � �. 2.
For Questions 3–6, perform the indicated matrix operations. If the matrix does not exist, write impossible.
3. � � � � � 3.
4. �16�� � � �
14�� � 4.
5. 3� � � 5� � 5.
6. � � � � � 6.
7. Use A � � �, B � � �, and C � � � to find 7.
C(BA) and (AB)C. Then state whether C(BA) � (AB)C is true for the given matrices.
8. Quadrilateral WXYZ with vertices W(�4, 2), X(�1, 4), 8.Y(6, �1), and Z(�2, �3) is translated so that W� is at (5, �2).Find the coordinates of X�, Y�, and Z�.
9. A triangle is rotated 270� counterclockwise about the origin. 9.The coordinates of the new vertices are R�(�4, 1), S�(1, �8),and T�(6, 4). What were the coordinates of the triangle in its original position?
10. Find the value of � �. 10.
11. Solve for x if � � � 18. 11.�4 2x3 x
6.0 �3.1�5.9 �4.8
0 �42 �1
�1 ��32��4 �3
2 2
4 �2 ��12�
6 1 0
�1 �12�
0 4
�1 3 �14�
�25� �1 0
��12� �1 0
3 �15� 2
8 21 �1
12 0�6 3
0.95 �7.243.59 3.412.68 �5.01
2.37 4.121.69 3.97
�5.18 6.25
3 x � 11 4
x2 � 1 4 � y2x � y y � 1
NAME DATE PERIOD
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Ass
essm
ent
© Glencoe/McGraw-Hill 228 Glencoe Algebra 2
Chapter 4 Test, Form 3 (continued)
12. Evaluate � � using expansion by minors. 12.
13. GEOMETRY Find the area of a triangle whose vertices are 13.
located at ��4, �12��, ���
52�, �1�, and (6, �2). Evaluate the
determinant using diagonals.
For Questions 14 and 15, use Cramer’s Rule to solve each system of equations.
14. �12�x � �
14�y � 1
3x � �12�y � �4
16. Determine whether M � � � and N � � � are 16.
inverses.
17. Find the inverse of A � � �, if it exists. 17.
For Questions 18 and 19, solve each system of equations by using inverse matrices.
18. 9a � 4b � �5 18.6a � 2b � �3
19. 5x � 3y � �4.5 19.2x � 1.2y � �1.8
20. BUDGET A foundation spent all of its operating budget 20.on administrative expenses, renewal grants, and low-interest loans. Administrative expenses and renewal grants accounted for 78.4% of the total budget. Renewable grants accounted for 5.1% more of the total budget than low-interest loans. Let a,r, and l each represent the percent of the budget accounted for by administrative expenses, renewal grants, and low-interest loans, respectively. Write a matrix equation that describes this situation.
Bonus Find the value of � �. B:a � 1 a 1b � 1 b �1c � 1 c 1
�15� �
23�
��25� ��
13�
�25� 1
�15� 1
5 �5�1 2
15. 3m � 4n � 6p � 152m � 3n � 5p � �115m � 6n � p � 9
7 5 4�3 �9 5
2 0 �3
NAME DATE PERIOD
44
14.
15.
Chapter 4 Open-Ended Assessment
© Glencoe/McGraw-Hill 229 Glencoe Algebra 2
Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solutions in more thanone way or investigate beyond the requirements of the problem.
1. State any conditions or restrictions on m, n, j, and k when performing the indicated operations for matrices Am�n and Bj�k.a. adding A and Bb. multiplying A and Bc. finding the determinant of Ad. multiplying B by a scalar ce. finding the inverse of A
2. Choose three points in the coordinate plane so that no two points lie in the same quadrant. Consider these points the vertices of �XYZ.a. Write the vertex matrix V for �XYZ, and state the dimensions of V.
For parts b through f, use T � � �, R � � �, C � � �,and I � � � to find each sum or product. Then state the type of
transformation represented by the new matrix, describing the relationship between the image and premimage of �XYZ.
b. T � V c. 3V d. RV e. CV f. IV
3. Suppose you are asked to find the value of a if � � � 0.
a. If you must evaluate the determinant above using expansion by minors,which row would you use for the expansion? Explain your reasoning.
b. If you were able to choose which method, either using expansion by minors or using diagonals, to evaluate this determinant, which method would you choose? Why?
c. Find the value of a using the method of your choice.
4. Christopher purchased three CDs and two videos, and spent a total of $85. Edward purchased two CDs and one video, spending $50.a. Write a system of equations that describes this situation. Explain
what the variables in your system represent.b. Use Cramer’s Rule to solve your system. Interpret the meaning of
your solution.c. Write a matrix equation for your system of equations. Then solve
the matrix equation by using inverse matrices.d. Which of the two methods do you prefer. Why?
2 �7 57 a �21 0 �1
1 00 1
0 �11 0
�1 00 1
3 3 3�4 �4 �4
NAME DATE PERIOD
SCORE 44
Ass
essm
ent
© Glencoe/McGraw-Hill 230 Glencoe Algebra 2
Chapter 4 Vocabulary Test/Review
Underline or circle the correct word or phrase to complete each sentence.
1. (Scalar multiplication, Translation, Cramer’s Rule) is a methodfor solving systems of equations that uses determinants.
2. A (row matrix, square matrix, column matrix) is a matrix withthe same number of rows as columns.
3. Any function that maps points of a preimage onto its image iscalled a(n) (transformation, isometry, dilation).
4. A rectangular array of variables or constants is called a(n)(scalar, matrix, element).
5. A (reflection, rotation, translation) is a transformation in whichevery point of a figure is mapped to a corresponding image acrossa line of symmetry.
6. (Cramer’s Rule, Scalar multiplication, Expansion by minors) is amethod for evaluating a third-order determinant.
7. If the row and column containing a specific element of a 3 � 3matrix are deleted, the resulting 2 � 2 matrix is called the(inverse, zero matrix, minor) of that element.
8. A transformation in which the preimage and the image arecongruent figures is called a(n) (isometry, scalar, dilation).
9. For triangle ABC, the coordinates of points A, B, and C can bewritten in a (row matrix, vertex matrix, column matrix).
10. A transformation that enlarges or reduces a figure is called a(translation, dilation, rotation).
In your own words—Define each term.
11. identity matrix
12. column matrix
column matrixCramer’s Ruledeterminantdilationdimensionelementequal matrices
expansion by minorsidentity matriximageinverseisometrymatrixmatrix equation
minorpreimagereflectionrotationrow matrixscalarscalar multiplication
second-order determinantsquare matrixthird-order determinanttransformationtranslationvertex matrixzero matrix
NAME DATE PERIOD
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Chapter 4 Quiz (Lessons 4–1 and 4–2)
44
© Glencoe/McGraw-Hill 231 Glencoe Algebra 2
1. GRADES On the first two algebra exams of the year, 1.Trista’s grades were 80 and 95, Javier’s grades were 85 and 90, and Yolanda’s grades were 75 and 90. Write a 3 � 2 matrix to organize this information.
2. Solve the matrix equation � � � � �. 2.
Use matrices A, B, and C to find the following.
A � � � B � � � C � � �
3. the dimensions of matrix A
4. B � C 5. 3C � 2B
1 7 �5�10 0 2
3 5 �812 �11 9
6 4 �8 51 �3 9 72 0 �2 �4
�29
x � 5y2x � 3y
NAME DATE PERIOD
SCORE
Chapter 4 Quiz (Lessons 4–3 and 4–4)
1. Determine whether the matrix product A5�2 � B2�4 is defined. 1.If so, state the dimensions of the product.
2. If possible, find the product � � � � �. 2.
3. Use A � � � and B � � �, and scalar c � 3 to find 3.
c(AB) and (BA)c. Then state whether the equation c(AB) � (BA)c is true for the given matrices.
4. Standardized Test Practice 4.Rectangle A�B�C�D� is the result of a translation of rectangle ABCD. A table of the translations is shown. Find the coordinates of A and D�.
5. The vertices of �XYZ are X(2, 3), Y(�3, 2), and Z(4, �5). The 5.triangle is being reflected over the line y � x. Find the coordinates of �X�Y�Z�.
�5 24 3
1 �24 3
2 �13 �31 4
1 5 �46 0 8
NAME DATE PERIOD
SCORE 44
3.
4.
5.
RectangleABCD
RectangleA�B�C�D�
A A� (1, 1)
B (1, �4) B� (4, 1)
C (1, 1) C� (4, 6)
D (�2, 1) D�
© Glencoe/McGraw-Hill 232 Glencoe Algebra 2
1. Find the value of � �. 1.
2. Evaluate � � using expansion by minors. 2.
3. GEOMETRY Find the area of a triangle whose vertices 3.are located at (�2, 5), (�4, �3), and (3, 1). Evaluate the determinant using diagonals.
Use Cramer’s Rule to solve each system of equations.
4. x � y � 6 4.3x � 2y � 11
5. 4x � 2y � z � �1 5.3x � 4y � 5z � �13x � 4y � 3z � 7
2 �1 34 0 1
�2 3 5
�3 42 �5
Chapter 4 Quiz (Lessons 4–7 and 4–8)
1. Determine whether A � � � and B � � � are 1.
inverses.
2. Find the inverse of M � � �, if it exists. 2.
For Questions 3 and 4, write a matrix equation for each system of equations.
3. a � 3b � 15 3.a � 4b � �13
4. 2x � 3y � 4z � �20 4.3x � z � 2x � 4y � z � �6
5. Solve the system of equations 3x � 2y � 22 and x � 2y � �6 5.by using inverse matrices.
�8 24 �1
�12� �
14�
0 ��12�
2 10 �2
NAME DATE PERIOD
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Chapter 4 Quiz (Lessons 4–5 and 4–6)
44
NAME DATE PERIOD
SCORE
44
Chapter 4 Mid-Chapter Test (Lessons 4–1 through 4–4)
© Glencoe/McGraw-Hill 233 Glencoe Algebra 2
Write the letter for the correct answer in the blank at the right of each question.
A. 16 � 8 B. 2 � 2 � 3 C. 4 � 3 D. 3 � 4 1.
2. Solve the matrix equation � � � � � for y.
A. �1 B. 7 C. �3 D. 3 2.
3. For all matrices X3�5, Y2�3, Z3�4, and scalars c, which statement is always true?A. c(YZ) � (YZ)c B. YX � XYC. Y � Z � Z � Y D. c(ZX) � c(XY) 3.
4. Triangle RST with vertices R(�3, 4), S(0, �5), and T(6, 7) is translated 3 units left and 5 units down. Find the coordinates of R�.A. (�6, 1) B. (�6, �1) C. (0, �1) D. (0, 9) 4.
5. Parallelogram ABCD with A(�2, �2), B(4, �2), C(5, 3), and D(�1, 3) is rotated 270� counterclockwise about the origin. Find the coordinates of B�.A. (2, 4) B. (�4, 2) C. (2, �4) D. (�2, �4) 5.
Perform the indicated matrix operations. If the matrix does not exist, writeimpossible.
6. � � � � � 6.
7. � � � � � 7.
8. �4� � 8.
9. 3[1 0 �9 6] � 4[5 12 0 �3] 9.
10. � � � � � 10.
11. � � � � � 11.4 6�2 7
�1 2 10 5 3
1 �23 �1
�4 01 5
3 �5 129 11 �7
�2 4 6
9 �7 �3 06 �5 1 19
0 6 13 �11�7 �4 2 4
4 �6 13 5�5 11 0 �4
3 12 0 79 �15 �8 6
Part II
117
3x � 2y4x � 5y
Part I
NAME DATE PERIOD
SCORE 44
Ass
essm
ent
1. State the dimensions of matrix F if F � � �.0 1 02 �4 24 �8 48 �16 8
© Glencoe/McGraw-Hill 234 Glencoe Algebra 2
Chapter 4 Cumulative Review (Chapters 1–4)
1. The formula S � �n(n
2� 1)� can be used to find the sum of the 1.
first n natural numbers. Find the sum of the first 100 natural numbers. (Lesson 1-1)
2. Evaluate 8 � � 3a � b � if a � 4.3 and b � �15. 2.(Lesson 1-4)
3. Find the slope of the line that passes through (4, –3) and 3.(–1, –7). (Lesson 2-3)
4. Identify the domain and range of the function g(x) � � x � �4. 4.(Lesson 2-6)
5. Graph y � 2x � � 1. (Lesson 2-7) 5.
6. The vertex of an angle is the point where the lines whose 6.equations are y � 2x and y � x � 1 meet. Find the coordinates of the vertex. (Lesson 3-1)
7. Determine whether the ordered pair (–2, 6) is a solution of 7.the system y 2x � 7 and y 5 � 3x. (Lesson 3-3)
8. Write a system of three equations to represent the given 8.information. The sum of three numbers is 37. The second number is 4 more than the first number, and the sum of the second number and the third number is 29. (Lesson 3-5)
9. Solve � � � � �. (Lesson 4-1) 9.
For Questions 10 and 11, perform the indicated matrix 10.operations. If the matrix does not exist, write impossible.
10. � � � � � 11. �4� � 11.
(Lesson 4-2) (Lesson 4-2)
12. Evaluate � � using diagonals. (Lesson 4-5) 12.
13. Find the inverse of M � � �, if it exists. (Lesson 4-7) 13.3 2�1 4
�4 2 �11 �1 2
�3 0 5
3 0 11�9 2 6
4 �3 �5
1 9 �5�7 6 4
3 1�2 17
�814
2x � yx � 4y
y
xO
NAME DATE PERIOD
44
Standardized Test Practice (Chapters 1–4)
© Glencoe/McGraw-Hill 235 Glencoe Algebra 2
1. If 4x2 � 13, then what is the value of (2x � 3)(2x � 3)?A. 4 B. 22 � 6�3� C. �13� � 9 D. –22 1.
2. If the measure of the edge of a cube is 3, the measure of the surface area of the cube is _____.E. 36 F. 54 G. 27 H. 9 2.
3. If the average of a and b equals the average of a, b, and c, then express c in terms of a and b.
A. a � b B. 2(a � b) C. �a �
2b
� D. �a �
3b
� 3.
4. There are 100 items in a garage sale. 30% of the items are sold during the first hour of the sale. If 10 items are sold during the second hour, what percent of the items have not been sold?E. 10% F. 60% G. 70% H. 90% 4.
5. If it takes 6 hours for 2 people to clean a house, how many hours will it take 4 people, working at the same rate, to clean another house that is the same size?
A. �23� B. 1�
12� C. 2 D. 3 5.
6. Which statement must be true?E. x y F. x � yG. x � y H. 2y � 180 � x 6.
7. Three less than three times a number is �56�. What is one more than
twice the number?
A. 2�23� B. 3�
49� C. 3�
59� D. 2�
12� 7.
8. r�s is defined as 2r2 � s2 � 4rs. What is the value of �2��1?E. –17 F. –1 G. 1 H. 17 8.
9. Which is closest to the value of �36.82.
�9
1.5�?
A. 19 B. 100 C. 150 D. 200 9.
10. In the figure, A�B� D�C�, DX � 8, and AD � 17.What is the length of BC?E. 17 F. 8�2�G. 15�2� H. 15 10. HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
NAME DATE PERIOD
44
Ass
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Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
2y˚(180 � x)˚
A B
D X CY
45˚
© Glencoe/McGraw-Hill 236 Glencoe Algebra 2
Standardized Test Practice (continued)
11. What is the value of r3 if �1r� � �0.01�? 11. 12.
12. If 5y�4 � 25 and 32x�1 � 27, what is the
value of �xy�?
13. For what integer value of m is 2m � 14 42
and 5 � �m4� � 3 � 7?
14. If D�C� is parallel to the 13. 14.x-axis and BC � 5 in the figure, what is the area of the shaded region?
Column A Column B
15. r 1 15.
16. a : b � c : d 16.
17. a 3, b � �4 17.
18. A � {7, 15, 2, 8, 14, 2} 18.
The mean of set A.The median of set A.
DCBA
�ba
� � 1�a �
ab
�
DCBA
bdac
DCBA
10r � 110r 2 � r
DCBA
Part 3: Quantitative Comparison
Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
NAME DATE PERIOD
44
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
y
xOC(10, 1)
A(�2, 5) B
D
A
D
C
B
Unit 1 Test(Chapters 1–4)
© Glencoe/McGraw-Hill 237 Glencoe Algebra 2
1. Find the value of 7 � 2(5 � 2 � 32). 1.
2. Name the sets of numbers to which 457 belongs. 2.
3. The sum of a number and 17 more than twice the same 3.number is 101. Find the number.
4. Evaluate 5 � � a � 5b � if a � �12 and b � 2. 4.
5. Define a variable and write an inequality. Then solve. 5.A local summer baseball team plays 20 games each season.So far, they have won 9 games and lost 2. How many more games must they win this season to win at least 75% of all their games?
6. Solve 3 � 2(1 � x) 6 or 2x � 14 8. Graph the solution set 6.on a number line.
7. Solve 3 � � 2y � 1 � � 1. Graph the solution set on a number 7.line.
8. If f(x) � �5x2
x� 4�, find f(4). 8.
9. Write an equation in slope-intercept form for the line that 9.passes through (3, 5) and (–2, 1).
10. Write an equation for the line that passes through (0, 7) and 10.
is perpendicular to the line whose equation is y � �12
�x � 1.
For Questions 11 and 12, use the set of data in the table.
The table shows the relationship between the price of a comic book and the number of copies sold. 11.
11. Draw a scatter plot for the data.
12. Use two ordered pairs to write a prediction equation. Then 12.use your prediction equation to predict the number of comic books sold when the price is $4.50.
13. Evaluate h���23
�� if h(x) � x � 2 �. 13.
14. Describe the system 6x � 2y � 10 and 9x � 3y � 8 as 14.consistent and independent, consistent and dependent, or inconsistent.
n
p
68
0
1012141618
1 2 3 4Price (dollars)
Num
ber
Sol
d
�4�5 �3 �2 �1 0 2 3 4 5 61
0 1�3 �2 �1�4
NAME DATE PERIOD
SCORE
Ass
essm
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Price p (in dollars) 2.00 2.50 2.75 3.00 3.50
Number sold n 16 13 12 10 7
© Glencoe/McGraw-Hill 238 Glencoe Algebra 2
Unit 1 Test (continued)(Chapters 1–4)
15. Solve the system of equations at 4x � 2y � 2 15.the right by using substitution. y � �4x � 7
16. Solve the system of equations at x � 3y � 5 16.the right by using elimination. 3x � y � 5
For Questions 17–19, use the following information.A furniture company displays bedroom sets which require 21 square meters of space and living room sets which require 42 square meters of space. The company, which has 546 square meters of available space, wants to display at least 6 bedroom sets and at least 5 living room sets. 17.
17. Let b represent the number of bedroom sets and � represent the number of living room sets. Write a system of inequalities to represent the number of furniture sets that can be displayed.
18. Draw the graph showing the feasible region. Label the 18.coordinates of the vertices of the feasible region.
19. If a bedroom set sells for $10,000 and a living room set sells 19.for $18,000, determine the number of bedroom sets and living room sets that must be sold to maximize the amount collected.
For Questions 20 and 21, use the matrices below.
A � � � B � � � C � � �20. Find A � B. 20.
21. Find BC, if possible. 21.
22. Triangle DEF with vertices D(2, 5), E(1, –6), and 22.F(–5, 3) is translated 3 units right and 2 units down. Find the coordinates of D�E�F�.
23. Evaluate � � using expansion by minors. 23.
24. Use Cramer’s Rule to solve the system of equations 24.3x � 5y � 21 and 4x � 2y � 2.
25. Solve the matrix equation � � � � � � � � 25.
using inverse matrices.
32�5
mn
4 �51 2
12 5 �2�3 0 1�5 4 2
�12
�2
10 6 �7�4 3 0
17 2 311 4 �9
y
xO
4
8
12
4 8 12 16
NAME DATE PERIOD
Standardized Test PracticeStudent Record Sheet (Use with pages 216–217 of the Student Edition.)
© Glencoe/McGraw-Hill A1 Glencoe Algebra 2
NAME DATE PERIOD
44
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7 9
2 5 8 10
3 6
Solve the problem and write your answer in the blank.
For Questions 13–17, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.
11 13 15 17
12
14 16
Select the best answer from the choices given and fill in the corresponding oval.
18 20 22
19 21 DCBADCBA
DCBADCBADCBA
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.
99 9 987654321
87654321
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0 0 0
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.
99 9 987654321
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DCBADCBA
DCBADCBADCBADCBA
DCBADCBADCBADCBA
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 3 Quantitative ComparisonPart 3 Quantitative Comparison
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 4-1)
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to
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cGra
w-H
ill16
9G
lenc
oe A
lgeb
ra 2
Lesson 4-1
Org
aniz
e D
ata
Mat
rix
a re
ctan
gula
r ar
ray
of v
aria
bles
or
cons
tant
s in
hor
izon
tal r
ows
and
vert
ical
col
umns
, us
ually
enc
lose
d in
bra
cket
s.
A m
atri
x ca
n b
e de
scri
bed
by i
ts d
imen
sion
s.A
mat
rix
wit
h m
row
s an
d n
colu
mn
s is
an
m
�n
mat
rix.
Ow
ls’ e
ggs
incu
bat
e fo
r 30
day
s an
d t
hei
r fl
edgl
ing
per
iod
is
also
30
day
s.S
wif
ts’ e
ggs
incu
bat
e fo
r 20
day
s an
d t
hei
r fl
edgl
ing
per
iod
is
44 d
ays.
Pig
eon
egg
s in
cub
ate
for
15 d
ays,
and
th
eir
fled
glin
g p
erio
d i
s 17
day
s.E
ggs
of t
he
kin
g p
engu
in i
ncu
bat
e fo
r 53
day
s,an
d t
he
fled
glin
g ti
me
for
a k
ing
pen
guin
is
360
day
s.W
rite
a 2
�4
mat
rix
to o
rgan
ize
this
in
form
atio
n.
Sour
ce: T
he C
ambr
idge
Fac
tfind
er
Wh
at a
re t
he
dim
ensi
ons
of m
atri
x A
if A
�?
Sin
ce m
atri
x A
has
2 r
ows
and
4 co
lum
ns,
the
dim
ensi
ons
of A
are
2 �
4.
Sta
te t
he
dim
ensi
ons
of e
ach
mat
rix.
1.4
�4
2.[1
612
0]1
�3
3.5
�2
4.A
tra
vel
agen
t pr
ovid
es f
or p
oten
tial
tra
vele
rs t
he
nor
mal
hig
h t
empe
ratu
res
for
the
mon
ths
of J
anu
ary,
Apr
il,J
uly
,an
d O
ctob
er f
or v
ario
us
citi
es.I
n B
osto
n t
hes
e fi
gure
s ar
e36
°,56
°,82
°,an
d 63
°.In
Dal
las
they
are
54°
,76°
,97°
,an
d 79
°.In
Los
An
gele
s th
ey a
re68
°,72
°,84
°,an
d 79
°.In
Sea
ttle
th
ey a
re 4
6°,5
8°,7
4°,a
nd
60°,
and
in S
t.L
ouis
th
ey a
re38
°,67
°,89
°,an
d 69
°.O
rgan
ize
this
inf
orm
atio
n in
a 4
�5
mat
rix.
Sour
ce: T
he N
ew Yo
rk T
imes
Alm
anac
Bo
sto
nD
alla
sL
os
An
gel
esS
eatt
leS
t.L
ou
isJa
nu
ary
36
5468
4638
A
pri
l
5676
7258
67
July
82
9784
7489
O
cto
ber
63
7979
6069
71
44
39
27
45
16
92
53
78
65
15
527
�4
23
60
5 1
470
24�
3 6
33
4290
13
10�
345
28
1580
Ow
lS
wift
Pig
eon
Kin
g P
engu
inIn
cuba
tion
30
2015
53
Fle
dglin
g
3044
1736
0
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill17
0G
lenc
oe A
lgeb
ra 2
Equ
atio
ns
Invo
lvin
g M
atri
ces
Eq
ual
Mat
rice
sTw
o m
atric
es a
re e
qual
if t
hey
have
the
sam
e di
men
sion
s an
d ea
ch e
lem
ent
of o
ne m
atrix
iseq
ual t
o th
e co
rres
pond
ing
elem
ent
of t
he o
ther
mat
rix.
You
can
use
th
e de
fin
itio
n o
f eq
ual
mat
rice
s to
sol
ve m
atri
x eq
uat
ion
s.
Sol
ve
�
for
xan
d y
.
Sin
ce t
he
mat
rice
s ar
e eq
ual
,th
e co
rres
pon
din
g el
emen
ts a
re e
qual
.Wh
en y
ou w
rite
th
ese
nte
nce
s to
sh
ow t
he
equ
alit
y,tw
o li
nea
r eq
uat
ion
s ar
e fo
rmed
.4x
��
2y�
2y
�x
�8
Th
is s
yste
m c
an b
e so
lved
usi
ng
subs
titu
tion
.4x
��
2y�
2F
irst
equa
tion
4x�
�2(
x�
8) �
2S
ubst
itute
x�
8 fo
r y.
4x�
�2x
�16
�2
Dis
trib
utiv
e P
rope
rty
6x�
18A
dd 2
xto
eac
h si
de.
x�
3D
ivid
e ea
ch s
ide
by 6
.
To
fin
d th
e va
lue
of y
,su
bsti
tute
3 f
or x
in e
ith
er e
quat
ion
.y
�x
�8
Sec
ond
equa
tion
y�
3 �
8S
ubst
itute
3 f
or x
.
y�
�5
Sub
trac
t.
Th
e so
luti
on i
s (3
,�5)
.
Sol
ve e
ach
eq
uat
ion
.
1.[5
x4y
] �
[20
20]
2.�
3.�
(4,5
)�
,�6 �
(�2,
�7)
4.�
5.�
6.�
(4,2
.5)
(6,�
3)(�
5,8)
7.�
8.�
9.�
�,�
8 ���
,��
(18,
21)
10.
�11
.�
12.
�
(2,5
.2)
(2.5
,4)
(�12
.5,�
12.5
)
0
�25
x
�y
x�
y
17
�8
2x
�3y
�
4x�
0.5y
7
.5
8.0
3
x�
1.5
2y
�2.
4
1 � 23 � 4
5 � 4
9
51
� 3x �
�� 7y �
� 2x ��
2y
�
3 �
11
8x
�6y
1
2x�
4y
18
20
12
�13
8
x�
y16
x
12y
�4x
�
1 �
18
5x
�3y
2x�
y
3 1
22
x�
3y
x
�2y
�
1 2
2
x�
2y
3x
�4y
4 � 3
4 �
5x
y
�5
�2y
x2
8 �
4y
�3x
�2
3x
y
�2y
� 2
x�
84
x
y
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Intr
od
uct
ion
to
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-1
4-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-1)
Skil
ls P
ract
ice
Intr
od
uct
ion
to
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-1
4-1
©G
lenc
oe/M
cGra
w-H
ill17
1G
lenc
oe A
lgeb
ra 2
Lesson 4-1
Sta
te t
he
dim
ensi
ons
of e
ach
mat
rix.
1.2
�3
2.[0
15]
1 �
2
3.2
�2
4.3
�3
5.2
�4
6.4
�1
Sol
ve e
ach
eq
uat
ion
.
7.[5
x3y
] �
[15
12]
(3,4
)8.
[3x
�2]
�[7
]3
9.�
(�2,
7)10
.[2x
�8y
z] �
[10
16�
1](5
,�2,
�1)
11.
�(4
,5)
12.
�(2
,4)
13.
�(�
4,3)
14.
�(0
,�2)
15.
�(1
,�1)
16.
�(2
,11,
7)
17.
�(9
,4,3
)18
.�
(1,4
,0)
19.
�(3
,1)
20.
�(1
2,3)
4y
x
� 3
x 3
y6
y
x
2x
y�
2
4x
�1
13
4z
5x
4
y�
3
8z
9
4y
9
x
16
3z
7
2y�
4 1
228
3
x�
118
124z
3x
y�
34
x�
1 9
y�
5
5x
�2
3y
�10
3
x�
2 7
y�
2�
20
8y
5
x 2
4
10x
32
20
5
6 �
6y
4
2
8
�x
2y
�8�
14
2y
7
x 1
4
�1
�1
�1
�3
93
�3
�6
34
�4
5
6
12
�3
45
�2
79
32
18
3
24
�1
40
©G
lenc
oe/M
cGra
w-H
ill17
2G
lenc
oe A
lgeb
ra 2
Sta
te t
he
dim
ensi
ons
of e
ach
mat
rix.
1.[�
3�
37]
1 �
32.
2 �
33.
3 �
4
Sol
ve e
ach
eq
uat
ion
.
4.[4
x42
] �
[24
6y]
(6,7
)
5.[�
2x22
�3z
] �
[6x
�2y
45]
(0,�
11,�
15)
6.�
(�6,
7)7.
�(4
,1)
8.�
(�3,
2)9.
�(�
1,1)
10.
�(�
1,�
1,1)
11.
�(7
,13)
12.
�(3
,�2)
13.
�(2
,�4)
14.T
ICK
ET P
RIC
EST
he
tabl
e at
th
e ri
ght
give
s
tick
et p
rice
s fo
r a
con
cert
.Wri
te a
2 �
3 m
atri
xth
at r
epre
sen
ts t
he
cost
of
a ti
cket
.
CO
NST
RU
CTI
ON
For
Exe
rcis
es 1
5 an
d 1
6,u
se t
he
foll
owin
g in
form
atio
n.
Du
rin
g ea
ch o
f th
e la
st t
hre
e w
eeks
,a
road
-bu
ildi
ng
crew
has
use
d th
ree
tru
ck-l
oads
of
gra
vel.
Th
e ta
ble
at t
he
righ
t sh
ows
the
amou
nt
of g
rave
l in
eac
h l
oad.
15.W
rite
a m
atri
x fo
r th
e am
oun
t of
gra
vel
in e
ach
loa
d.
or
16.W
hat
are
th
e di
men
sion
s of
th
e m
atri
x?3
�3
40
4032
3
240
24
24
3224
40
3224
4
040
32
32
2424
Wee
k 1
Wee
k 2
Wee
k 3
Load
140
ton
sLo
ad 1
40 t
ons
Load
132
ton
s
Load
232
ton
sLo
ad 2
40 t
ons
Load
224
ton
s
Load
324
ton
sLo
ad 3
32 t
ons
Load
324
ton
s
612
18
815
22
Ch
ildS
tud
ent
Ad
ult
Co
st P
urc
has
edin
Ad
van
ce$6
$12
$18
Co
st P
urc
has
edat
th
e D
oo
r$8
$15
$22
0
�2
2x
�y
3x
�2y
�
1
y5
x�
8y
3x
�11
y�
8
17
3x
y
�4
�5
x�
1 3
y5z
�4
�
53x
�1
2y
�1
3z�
2
�3x
�21
8y�
56
x�
12
�3y
�6
�
3x
�2y
�16
�
4x�
3
6y
20
2
y�
37
x�
8 8
y�
3�
36
17
6x
2y
�3
�2
2�
23
5
160
0
47
�1
4
58
�1
�2
18
Pra
ctic
e (
Ave
rag
e)
Intr
od
uct
ion
to
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-1
4-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 4-1)
Readin
g t
o L
earn
Math
em
ati
csIn
tro
du
ctio
n t
o M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-1
4-1
©G
lenc
oe/M
cGra
w-H
ill17
3G
lenc
oe A
lgeb
ra 2
Lesson 4-1
Pre-
Act
ivit
yH
ow a
re m
atri
ces
use
d t
o m
ake
dec
isio
ns?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-1
at
the
top
of p
age
154
in y
our
text
book
.
Wh
at i
s th
e ba
se p
rice
of
a M
id-S
ize
SU
V?
$27,
975
Rea
din
g t
he
Less
on
1.G
ive
the
dim
ensi
ons
of e
ach
mat
rix.
a.2
�3
b.[
14
0�
82]
1 �
5
2.Id
enti
fy e
ach
mat
rix
wit
h a
s m
any
of t
he
foll
owin
g de
scri
ptio
ns
that
app
ly:r
ow m
atri
x,co
lum
n m
atri
x,sq
uar
e m
atri
x,ze
ro m
atri
x.
a.[6
54
3]ro
w m
atri
x
b.
colu
mn
mat
rix;
zero
mat
rix
c.[0
]ro
w m
atri
x;co
lum
n m
atri
x;sq
uar
e m
atri
x;ze
ro m
atri
x
3.W
rite
a s
yste
m o
f eq
uat
ion
s th
at y
ou c
ould
use
to
solv
e th
e fo
llow
ing
mat
rix
equ
atio
n f
orx,
y,an
d z.
(Do
not
act
ual
ly s
olve
th
e sy
stem
.)
�3x
��
9,x
�y
�5,
y�
z�
6
Hel
pin
g Y
ou
Rem
emb
er
4.S
ome
stu
den
ts h
ave
trou
ble
rem
embe
rin
g w
hic
h n
um
ber
com
es f
irst
in
wri
tin
g th
edi
men
sion
s of
a m
atri
x.T
hin
k of
an
eas
y w
ay t
o re
mem
ber
this
.
Sam
ple
an
swer
:R
ead
th
e m
atri
x fr
om
to
p t
o b
ott
om
,th
en f
rom
left
to
rig
ht.
Rea
din
g d
ow
n g
ives
th
e n
um
ber
of
row
s,w
hic
h is
wri
tten
fir
st in
the
dim
ensi
on
s o
f th
e m
atri
x.R
ead
ing
acr
oss
giv
es t
he
nu
mb
er o
fco
lum
ns,
wh
ich
is w
ritt
en s
eco
nd
.
�9
5
6
3x
x
�y
y�
z
0
0
0
3
25
�1
06
©G
lenc
oe/M
cGra
w-H
ill17
4G
lenc
oe A
lgeb
ra 2
Tess
ella
tio
ns
A t
esse
llat
ion
is a
n a
ran
gem
ent
of p
olyg
ons
cove
rin
g a
plan
e w
ith
out
any
gaps
or
over
lapp
ing.
On
e ex
ampl
e of
a t
esse
llat
ion
is
a h
oney
com
b.T
hre
e co
ngr
uen
t re
gula
r h
exag
ons
mee
t at
ea
ch v
erte
x,an
d th
ere
is n
o w
aste
d sp
ace
betw
een
cel
ls.T
his
tess
ella
tion
is
call
ed a
reg
ula
r te
ssel
lati
on s
ince
it
is f
orm
ed
by c
ongr
uen
t re
gula
r po
lygo
ns.
A s
emi-
regu
lar
tess
ella
tion
is a
tes
sell
atio
n f
orm
ed b
y tw
o or
m
ore
regu
lar
poly
gon
s su
ch t
hat
th
e n
um
ber
of s
ides
of
the
poly
gon
s m
eeti
ng
at e
ach
ver
tex
is t
he
sam
e.
For
exa
mpl
e,th
e te
ssel
lati
on a
t th
e le
ft h
as t
wo
regu
lar
dode
cago
ns
and
one
equ
ilat
eral
tri
angl
e m
eeti
ng
at e
ach
vert
ex.W
e ca
n n
ame
this
tes
sell
atio
n a
3-1
2-12
for
th
en
um
ber
of s
ides
of
each
pol
ygon
th
at m
eet
at o
ne
vert
ex.
Nam
e ea
ch s
emi-
regu
lar
tess
ella
tion
sh
own
acc
ord
ing
to t
he
nu
mb
er
of s
ides
of
the
pol
ygon
s th
at m
eet
at e
ach
ver
tex.
1.2.
3-3-
3-3-
64-
6-12
An
eq
uil
ater
al t
rian
gle,
two
squ
ares
,an
d a
reg
ula
r h
exag
on c
an b
e u
sed
to
surr
oun
d a
poi
nt
in t
wo
dif
fere
nt
ord
ers.
Con
tin
ue
each
p
atte
rn t
o se
e w
hic
h i
s a
sem
i-re
gula
r te
ssel
lati
on.
3.3-
4-4-
64.
3-4-
6-4
no
t se
mi-
reg
ula
rse
mi-
reg
ula
r
On
an
oth
er s
hee
t of
pap
er,d
raw
par
t of
eac
h d
esig
n.T
hen
det
erm
ine
if i
t is
a s
emi-
regu
lar
tess
ella
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.
5.3-
3-4-
126.
3-4-
3-12
7.4-
8-8
8.3-
3-3-
4-4
no
t se
mi-
reg
ula
rn
ot
sem
i-re
gu
lar
sem
i-re
gu
lar
sem
i-re
gu
lar
En
rich
men
t
NA
ME
____
____
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____
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AT
E__
____
____
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ER
IOD
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_
4-1
4-1
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-2)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Op
erat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
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____
____
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AT
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____
____
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ER
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4-2
4-2
©G
lenc
oe/M
cGra
w-H
ill17
5G
lenc
oe A
lgeb
ra 2
Lesson 4-2
Ad
d a
nd
Su
btr
act
Mat
rice
s
Ad
dit
ion
of
Mat
rice
s�
�
Su
btr
acti
on
of
Mat
rice
s�
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Fin
d A
�B
if A
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and
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.
A�
B�
�
� �
Fin
d A
�B
if A
�
and
B�
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A�
B�
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�
�
Per
form
th
e in
dic
ated
op
erat
ion
s.If
th
e m
atri
x d
oes
not
exi
st,w
rite
im
pos
sibl
e.
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2.�
3.�
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ssib
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45
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131
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Exam
ple1
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ple1
Exam
ple2
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ple2
Exer
cises
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cises
Scal
ar M
ult
iplic
atio
nYo
u c
an m
ult
iply
an
m�
nm
atri
x by
a s
cala
r k.
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lar
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ltip
licat
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k�
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B�
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d 3
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2A.
3B�
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ubst
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ultip
ly.
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ify.
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impl
ify.
Per
form
th
e in
dic
ated
mat
rix
oper
atio
ns.
If t
he
mat
rix
doe
s n
ot e
xist
,wri
teim
pos
sibl
e.
1.6
2.�
3.0.
2
4.3
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7
91
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28
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181
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cGra
w-H
ill17
6G
lenc
oe A
lgeb
ra 2
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Op
erat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
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AT
E__
____
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ER
IOD
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_
4-2
4-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 4-2)
Skil
ls P
ract
ice
Op
erat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
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AT
E__
____
____
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ER
IOD
____
_
4-2
4-2
©G
lenc
oe/M
cGra
w-H
ill17
7G
lenc
oe A
lgeb
ra 2
Lesson 4-2
Per
form
th
e in
dic
ated
mat
rix
oper
atio
ns.
If t
he
mat
rix
doe
s n
ot e
xist
,wri
teim
pos
sibl
e.
1.[5
�4]
�[4
5][9
1]2.
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3.[3
16]
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94
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7][�
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Use
A�
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d C
�
to f
ind
th
e fo
llow
ing.
11.A
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©G
lenc
oe/M
cGra
w-H
ill17
8G
lenc
oe A
lgeb
ra 2
Per
form
th
e in
dic
ated
mat
rix
oper
atio
ns.
If t
he
mat
rix
doe
s n
ot e
xist
,wri
teim
pos
sibl
e.
1.�
2.
�
3.�
3�
44.
7�
2
5.�
2�
4�
6.
�
Use
A�
,B�
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d C
�
to f
ind
th
e fo
llow
ing.
7.A
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.4B
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11.�
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3C12
.A�
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ECO
NO
MIC
SF
or E
xerc
ises
13
an
d 1
4,u
se t
he
tab
le t
hat
sh
ows
loan
s b
y an
eco
nom
icd
evel
opm
ent
boa
rd t
o w
omen
an
d m
en s
tart
ing
new
bu
sin
esse
s.
13.W
rite
tw
o m
atri
ces
that
re
pres
ent
the
nu
mbe
r of
new
busi
nes
ses
and
loan
am
oun
ts,
one
for
wom
en a
nd
one
for
men
.
,
14.F
ind
the
sum
of
the
nu
mbe
rs o
f n
ew b
usi
nes
ses
and
loan
am
oun
ts
for
both
men
an
d w
omen
ove
r th
e th
ree-
year
per
iod
expr
esse
d as
a
mat
rix.
15. P
ET N
UTR
ITIO
NU
se t
he
tabl
e th
at g
ives
nu
trit
ion
al
info
rmat
ion
for
tw
o ty
pes
of d
og f
ood.
Fin
d th
edi
ffer
ence
in
th
e pe
rcen
t of
pro
tein
,fat
,an
d fi
ber
betw
een
Mix
B a
nd
Mix
A e
xpre
ssed
as
a m
atri
x.[2
�4
3]
% P
rote
in%
Fat
% F
iber
Mix
A22
125
Mix
B24
88
63
1,43
1,00
07
31,
574,
000
63
1,33
9,00
0
36
864,
000
32
672,
000
28
562,
000
27
567,
000
41
902,
000
35
777,
000
Wo
men
Men
Bu
sin
esse
sL
oan
Bu
sin
esse
sL
oan
A
mo
un
t ($
)A
mo
un
t ($
)
1999
27$5
67,0
0036
$864
,000
2000
41$9
02,0
0032
$672
,000
2001
35$7
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0028
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9
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24
32
43
27
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e (
Ave
rag
e)
Op
erat
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s w
ith
Mat
rice
s
NA
ME
____
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AT
E__
____
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ER
IOD
____
_
4-2
4-2
16
�15
62
14
4575
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-2)
Readin
g t
o L
earn
Math
em
ati
csO
per
atio
ns
wit
h M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-2
4-2
©G
lenc
oe/M
cGra
w-H
ill17
9G
lenc
oe A
lgeb
ra 2
Lesson 4-2
Pre-
Act
ivit
yH
ow c
an m
atri
ces
be
use
d t
o ca
lcu
late
dai
ly d
ieta
ry n
eed
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-2
at
the
top
of p
age
160
in y
our
text
book
.
•W
rite
a s
um
th
at r
epre
sen
ts t
he
tota
l n
um
ber
of C
alor
ies
in t
he
pati
ent’s
diet
for
Day
2.(
Do
not
act
ual
ly c
alcu
late
th
e su
m.)
482
�62
2 �
987
•W
rite
th
e su
m t
hat
rep
rese
nts
th
e to
tal
fat
con
ten
t in
th
e pa
tien
t’s d
iet
for
Day
3.(
Do
not
act
ual
ly c
alcu
late
th
e su
m.)
11 �
12 �
38
Rea
din
g t
he
Less
on
1.F
or e
ach
pai
r of
mat
rice
s,gi
ve t
he
dim
ensi
ons
of t
he
indi
cate
d su
m,d
iffe
ren
ce,o
r sc
alar
prod
uct.
If t
he in
dica
ted
sum
,dif
fere
nce,
or s
cala
r pr
oduc
t do
es n
ot e
xist
,wri
te i
mpo
ssib
le.
A�
B�
C�
D�
A�
D:
C�
D:
5B:
�4C
:2D
�3A
:
2.S
upp
ose
that
M,N
,an
d P
are
non
zero
2 �
4 m
atri
ces
and
kis
a n
egat
ive
real
nu
mbe
r.In
dica
te w
het
her
eac
h o
f th
e fo
llow
ing
stat
emen
ts i
s tr
ue
or f
alse
.
a.M
�(N
�P
) �
M�
(P�
N) tr
ue
b.
M�
N�
N�
Mfa
lse
c.M
�(N
�P
) �
(M�
N)
�P
fals
ed
.k
(M�
N)
�kM
�kN
tru
e
Hel
pin
g Y
ou
Rem
emb
er
3.T
he
mat
hem
atic
al t
erm
sca
lar
may
be
un
fam
ilia
r,bu
t it
s m
ean
ing
is r
elat
ed t
o th
e w
ord
scal
eas
use
d in
a s
cale
of
mil
eson
a m
ap.H
ow c
an t
his
usa
ge o
f th
e w
ord
scal
eh
elp
you
rem
embe
r th
e m
ean
ing
of s
cala
r?
Sam
ple
an
swer
:A
sca
le o
f m
iles
tells
yo
u h
ow
to
mu
ltip
ly t
he
dis
tan
ces
you
mea
sure
on
a m
ap b
y a
cert
ain
nu
mb
er t
o g
et t
he
actu
al d
ista
nce
bet
wee
n t
wo
loca
tio
ns.
Th
is m
ult
iplie
r is
oft
en c
alle
d a
sca
lefa
cto
r.A
scal
arre
pre
sen
ts t
he
sam
e id
ea:
It is
a r
eal n
um
ber
by
wh
ich
a m
atri
xca
n b
e m
ult
iplie
d t
o c
han
ge
all t
he
elem
ents
of
the
mat
rix
by a
un
iform
scal
efa
cto
r.
2 �
33
�2
2 �
2im
po
ssib
le2
�3
�3
60
�8
40
5
10
�3
6
412
�4
0
0�
5
35
6 �
28
1
©G
lenc
oe/M
cGra
w-H
ill18
0G
lenc
oe A
lgeb
ra 2
Su
nd
aram
’s S
ieve
Th
e pr
oper
ties
an
d pa
tter
ns
of p
rim
e n
um
bers
hav
e fa
scin
ated
man
y m
ath
emat
icia
ns.
In 1
934,
a yo
un
g E
ast
Indi
an s
tude
nt
nam
ed S
un
dara
m c
onst
ruct
ed t
he
foll
owin
g m
atri
x.
47
1013
1619
2225
..
.7
1217
2227
3237
42.
..
1017
2431
3845
5259
..
.13
2231
4049
5867
76.
..
1627
3849
6071
8293
..
..
..
..
..
..
..
A s
urp
risi
ng
prop
erty
of
this
mat
rix
is t
hat
it
can
be
use
d to
det
erm
ine
wh
eth
er o
r n
ot s
ome
nu
mbe
rs a
re p
rim
e.
Com
ple
te t
hes
e p
rob
lem
s to
dis
cove
r th
is p
rop
erty
.
1.T
he
firs
t ro
w a
nd
the
firs
t co
lum
n a
re c
reat
ed b
y u
sin
g an
ari
thm
etic
pa
tter
n.W
hat
is
the
com
mon
dif
fere
nce
use
d in
th
e pa
tter
n?
3
2.F
ind
the
nex
t fo
ur
nu
mbe
rs i
n t
he
firs
t ro
w.
28,3
1,34
,37
3.W
hat
are
th
e co
mm
on d
iffe
ren
ces
use
d to
cre
ate
the
patt
ern
s in
row
s 2,
3,4,
and
5?5,
7,9,
11
4.W
rite
th
e n
ext
two
row
s of
th
e m
atri
x.In
clu
de e
igh
t n
um
bers
in
eac
h r
ow.
row
6:
19,3
2,45
,58,
71,8
4,97
,110
;ro
w 7
:22
,37,
52,6
7,82
,97,
112,
127
5.C
hoo
se a
ny
five
nu
mbe
rs f
rom
th
e m
atri
x.F
or e
ach
nu
mbe
r n
,th
at y
ou
chos
e fr
om t
he
mat
rix,
fin
d 2n
�1.
An
swer
s w
ill v
ary.
6.W
rite
th
e fa
ctor
izat
ion
of
each
val
ue
of 2
n�
1 th
at y
ou f
oun
d in
pro
blem
5.
An
swer
s w
ill v
ary,
but
all n
um
ber
s ar
e co
mp
osi
te.
7.U
se y
our
resu
lts
from
pro
blem
s 5
and
6 to
com
plet
e th
is s
tate
men
t:If
n
occu
rs i
n t
he
mat
rix,
then
2n
�1
(is/
is n
ot)
a pr
ime
nu
mbe
r.
8.C
hoo
se a
ny
five
nu
mbe
rs t
hat
are
not
in
th
e m
atri
x.F
ind
2n�
1 fo
r ea
ch
of t
hes
e n
um
bers
.Sh
ow t
hat
eac
h r
esu
lt i
s a
prim
e n
um
ber.
An
swer
s w
ill v
ary,
but
all n
um
ber
s ar
e p
rim
e.
9.C
ompl
ete
this
sta
tem
ent:
If n
does
not
occ
ur
in t
he
mat
rix,
then
2n
�1
is
.a
pri
me
nu
mb
er
is n
ot
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-2
4-2
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 4-3)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Mu
ltip
lyin
g M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill18
1G
lenc
oe A
lgeb
ra 2
Lesson 4-3
Mu
ltip
ly M
atri
ces
You
can
mu
ltip
ly t
wo
mat
rice
s if
an
d on
ly i
f th
e n
um
ber
of c
olu
mn
sin
th
e fi
rst
mat
rix
is e
qual
to
the
nu
mbe
r of
row
s in
th
e se
con
d m
atri
x.
Mu
ltip
licat
ion
of
Mat
rice
s�
�
Fin
d A
Bif
A�
and
B�
.
AB
�
� S
ubst
itutio
n
�M
ultip
ly c
olum
ns b
y ro
ws.
�S
impl
ify.
Fin
d e
ach
pro
du
ct,i
f p
ossi
ble
.
1.�
2.�
3.�
4.�
5.�
6.�
7.�
[04
�3]
8.�
9.�
no
t p
oss
ible
10
�16
1
8�
6
59
11
�21
1
3�
15
2
�2
3
1 �
24
2
0�
3
14
�2
�1
31
1
�3
�2
07
�2
5�
4
610
�
43
�2
7
24
�15
1
4�
17
�1
4
84
�3
�9
�15
17
26
�2
�12
4
�1
�2
55
�2
2�
31
2 2
1
3�
2
04
�5
1
40
�2
�3
11
�3
1
5�
2
7
�7
14
14
�3
�2
2
34
12
3�
69
3�
1 2
43
�1
24
3
2 �
14
�1
0
37
30
03
4
1 �
23
�23
17
12
�10
�2
19
�
4(5)
�3(
�1)
�4(
�2)
�3(
3)
2(5
) �
(�2)
(�1)
2(�
2) �
(�2)
(3)
1(
5) �
7(�
1)1(
�2)
�7(
3)
5
�2
�1
3�
43
2
�2
1
7
5
�2
�1
3�
43
2�
2
1
7a
1x1
�b 1
x 2a 1
y 1�
b 1y 2
a
2x1
�b 2
x 2a 2
y 1�
b 2y 2
x
1y 1
x
2y 2
a
1b 1
a
2b 2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill18
2G
lenc
oe A
lgeb
ra 2
Mu
ltip
licat
ive
Pro
per
ties
Th
e C
omm
uta
tive
Pro
pert
y of
Mu
ltip
lica
tion
doe
s n
oth
old
for
mat
rice
s.
Pro
per
ties
of
Mat
rix
Mu
ltip
licat
ion
For
any
mat
rices
A,
B,
and
Cfo
r w
hich
the
mat
rix p
rodu
ct is
defin
ed,
and
any
scal
ar c
, th
e fo
llow
ing
prop
ertie
s ar
e tr
ue.
Ass
oci
ativ
e P
rop
erty
of
Mat
rix
Mu
ltip
licat
ion
(AB
)C�
A(B
C)
Ass
oci
ativ
e P
rop
erty
of
Sca
lar
Mu
ltip
licat
ion
c(A
B)
�(c
A)B
�A
(cB
)
Lef
t D
istr
ibu
tive
Pro
per
tyC
(A�
B)
�C
A�
CB
Rig
ht
Dis
trib
uti
ve P
rop
erty
(A�
B)C
�A
C�
BC
Use
A�
,B�
,an
d C
�
to f
ind
eac
h p
rod
uct
.
a.(A
�B
)C
(A�
B)C
��
���
�
�
� �
b.
AC
�B
C
AC
�B
C�
� �
�
��
��
�
Not
e th
at a
lth
ough
th
e re
sult
s in
th
e ex
ampl
e il
lust
rate
th
e R
igh
t D
istr
ibu
tive
Pro
pert
y,th
ey d
o n
ot p
rove
it.
Use
A�
,B�
,C�
,an
d s
cala
r c
��
4 to
det
erm
ine
wh
eth
er
each
of
the
foll
owin
g eq
uat
ion
s is
tru
e fo
r th
e gi
ven
mat
rice
s.
1.c(
AB
) �
(cA
)Bye
s2.
AB
�B
An
o
3.B
C�
CB
no
4.(A
B)C
�A
(BC
)ye
s
5.C
(A�
B)
�A
C�
BC
no
6.c(
A�
B)
�cA
�cB
yes
��1 2�
�2
1
�3
64
21
32
5�
2
�12
�21
�5
�20
2�
4 �
13�
19
�14
�17
8�
1
2(
1) �
0(6)
2(�
2) �
0(3)
5
(1)
�(�
3)(6
)5(
�2)
�(�
3)(3
)4
(1)
�( �
3)(6
)4(
�2)
�( �
3)(3
)
2(1)
�1(
6)2(
�2)
�1(
3)
1�
2 6
32
0 5
�3
1�
2 6
34
�3
21
�12
�21
�5
�20
6(1
) �
( �3)
(6)
6(�
2) �
( �3)
(3)
7(1
) �
(�2)
(6)
7(�
2) �
(�2)
(3)
1�
2 6
36
�3
7�
2
1�
2 6
32
0 5
�3
4�
3 2
1
1�
2 6
32
0 5
�3
4�
3 2
1
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Mu
ltip
lyin
g M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-3)
Skil
ls P
ract
ice
Mu
ltip
lyin
g M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill18
3G
lenc
oe A
lgeb
ra 2
Lesson 4-3
Det
erm
ine
wh
eth
er e
ach
mat
rix
pro
du
ct i
s d
efin
ed.I
f so
,sta
te t
he
dim
ensi
ons
ofth
e p
rod
uct
.
1.A
2 �
5�
B5
�1
2 �
12.
M1
�3
�N
3 �
21
�2
3.B
3 �
2�
A3
�2
un
def
ined
4.R
4 �
4�
S4
�1
4 �
1
5.X
3 �
3�
Y3
�4
3 �
46.
A6
�4
�B
4 �
56
�5
Fin
d e
ach
pro
du
ct,i
f p
ossi
ble
.
7.[3
2] �
[8]
8.�
9.�
10.
�n
ot
po
ssib
le
11.[
�3
4] �
[811
]12
.�
[2�
3�
2]
13.
�n
ot
po
ssib
le14
.�
15.
�16
.�
Use
A�
,B�
,C�
,a
nd
sca
lar
c�
2 to
det
erm
ine
wh
eth
er t
he
foll
owin
g eq
uat
ion
s ar
e tr
ue
for
the
give
n m
atri
ces.
17.(
AC
)c�
A(C
c)ye
s18
.AB
�B
An
o
19.B
(A�
C)
�A
B�
BC
no
20.(
A�
B)c
�A
c�
Bc
yes
3�
1 1
0�
32
5
12
1 2
1
4
4
2
2
2
01
1 1
10
�12
20
�
68
6
03
�3
02
�4
4 �
21
2
3
�6
61
512
3�
90
3 3
0
2�
2
45
�3
14
8
5
6 �
3
�2
32
6
�9
�6
�1
3
0�
1 2
2
1
3 �
11
3
�2
�3
�5
3
�2
1
3 �
11
28
�19
7�
92
�5
31
56
21
2
1
©G
lenc
oe/M
cGra
w-H
ill18
4G
lenc
oe A
lgeb
ra 2
Det
erm
ine
wh
eth
er e
ach
mat
rix
pro
du
ct i
s d
efin
ed.I
f so
,sta
te t
he
dim
ensi
ons
ofth
e p
rod
uct
.
1.A
7 �
4�
B4
�3
7 �
32.
A3
�5
�M
5 �
83
�8
3.M
2 �
1�
A1
�6
2 �
6
4.M
3 �
2�
A3
�2
un
def
ined
5.P
1 �
9�
Q9
�1
1 �
16.
P9
�1
�Q
1 �
99
�9
Fin
d e
ach
pro
du
ct,i
f p
ossi
ble
.
7.�
8.�
9.�
10.
�n
ot
po
ssib
le
11.[
40
2] �
[2]
12.
�[4
02]
13.
�14
.[�
15�
9] �
[�29
7�
75]
Use
A�
,B�
,C�
,an
d s
cala
r c
�3
to d
eter
min
e w
het
her
th
e fo
llow
ing
equ
atio
ns
are
tru
e fo
r th
e gi
ven
mat
rice
s.
15.A
C�
CA
yes
16.A
(B�
C)
�B
A�
CA
no
17.(
AB
)c�
c(A
B)
yes
18.(
A�
C)B
�B
(A�
C)
no
REN
TALS
For
Exe
rcis
es 1
9–21
,use
th
e fo
llow
ing
info
rmat
ion
.F
or t
hei
r on
e-w
eek
vaca
tion
,th
e M
onto
yas
can
ren
t a
2-be
droo
m c
ondo
min
ium
for
$179
6,a
3-be
droo
m c
ondo
min
ium
for
$2
165,
or a
4-b
edro
om c
ondo
min
ium
for
$253
8.T
he
tabl
e sh
ows
the
nu
mbe
r of
u
nit
s in
eac
h o
f th
ree
com
plex
es.
19.W
rite
a m
atri
x th
at r
epre
sen
ts t
he
nu
mbe
r of
eac
h t
ype
of u
nit
ava
ilab
le a
t ea
ch c
ompl
ex a
nd
a m
atri
x th
at
,re
pres
ents
th
e w
eekl
y ch
arge
for
eac
h t
ype
of u
nit
.
20.I
f al
l of
th
e u
nit
s in
th
e th
ree
com
plex
es a
re r
ente
d fo
r th
e w
eek
at t
he
rate
s gi
ven
th
e M
onto
yas,
expr
ess
the
inco
me
of e
ach
of
the
thre
e co
mpl
exes
as
a m
atri
x.
21.W
hat
is
the
tota
l in
com
e of
all
th
ree
com
plex
es f
or t
he
wee
k?$5
26,0
54
172
,452
2
27,9
60
125
,642
$17
96
$21
65
$25
38
36
2422
2
932
42
18
2218
2-B
edro
om
3-B
edro
om
4-B
edro
om
Su
n H
aven
3624
22
Su
rfsi
de
2932
42
Sea
bre
eze
1822
18
�1
0
0�
1
40
�2
�1
13
31
6
11
23
�10
�
3010
15�
55
0 0
5�
62
3
�1
4
02
12
06
�4
0�
2
1
3
�1
1
3
�1
3�
27
60
�5
3�
27
60
�5
�6
�12
3
93
24
7�
1�
30
2
5
2
20
�23
�5
�3
0
25
24
7�
13
0�
4�
6
3�
626
3
�2
7 6
0�
52
4 3
�1
Pra
ctic
e (
Ave
rag
e)
Mu
ltip
lyin
g M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 4-3)
Readin
g t
o L
earn
Math
em
ati
csM
ult
iply
ing
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill18
5G
lenc
oe A
lgeb
ra 2
Lesson 4-3
Pre-
Act
ivit
yH
ow c
an m
atri
ces
be
use
d i
n s
por
ts s
tati
stic
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-3
at
the
top
of p
age
167
in y
our
text
book
.
Wri
te a
su
m t
hat
sh
ows
the
tota
l po
ints
sco
red
by t
he
Oak
lan
d R
aide
rsdu
rin
g th
e 20
00 s
easo
n.(
Th
e su
m w
ill
incl
ude
mu
ltip
lica
tion
s.D
o n
otac
tual
ly c
alcu
late
th
is s
um
.)
6 �
58 �
1 �
56 �
3 �
23 �
2 �
1 �
2 �
2
Rea
din
g t
he
Less
on
1.D
eter
min
e w
het
her
eac
h i
ndi
cate
d m
atri
x pr
odu
ct i
s de
fin
ed.I
f so
,sta
te t
he
dim
ensi
ons
of t
he
prod
uct
.If
not
,wri
te u
nd
efin
ed.
a.M
3 �
2an
d N
2 �
3M
N:
NM
:
b.
M1
�2
and
N1
�2
MN
:N
M:
c.M
4 �
1an
d N
1 �
4M
N:
NM
:
d.
M3
�4
and
N4
�4
MN
:N
M:
2.T
he
regi
onal
sal
es m
anag
er f
or a
ch
ain
of
com
pute
r st
ores
wan
ts t
o co
mpa
re t
he
reve
nu
efr
om s
ales
of
one
mod
el o
f n
oteb
ook
com
pute
r an
d on
e m
odel
of
prin
ter
for
thre
e st
ores
in h
is a
rea.
Th
e n
oteb
ook
com
pute
r se
lls
for
$185
0 an
d th
e pr
inte
r fo
r $1
75.T
he
nu
mbe
rof
com
pute
rs a
nd
prin
ters
sol
d at
th
e th
ree
stor
es d
uri
ng
Sep
tem
ber
are
show
n i
n t
he
foll
owin
g ta
ble.
Wri
te a
mat
rix
prod
uct
th
at t
he
man
ager
cou
ld u
se t
o fi
nd
the
tota
l re
ven
ue
for
com
pute
rs a
nd
prin
ters
for
eac
h o
f th
e th
ree
stor
es.(
Do
not
cal
cula
te t
he
prod
uct
.)
�
Hel
pin
g Y
ou
Rem
emb
er
3.M
any
stu
den
ts f
ind
the
proc
edu
re o
f m
atri
x m
ult
ipli
cati
on c
onfu
sin
g at
fir
st b
ecau
se i
tis
un
fam
ilia
r.T
hin
k of
an
eas
y w
ay t
o u
se t
he
lett
ers
R a
nd
C t
o re
mem
ber
how
to
mu
ltip
ly m
atri
ces
and
wh
at t
he
dim
ensi
ons
of t
he
prod
uct
wil
l be
.S
amp
le a
nsw
er:
Just
rem
emb
er R
C f
or
“ro
w,c
olu
mn
.”M
ult
iply
eac
h r
owo
f th
e fi
rst
mat
rix
by e
ach
co
lum
no
f th
e se
con
d m
atri
x.T
he
dim
ensi
on
s o
f th
e p
rod
uct
are
the
nu
mb
er o
f ro
ws
of
the
firs
t m
atri
x an
d t
he
nu
mb
er o
f co
lum
ns
of
the
seco
nd
mat
rix.1
850
17
51
2810
12
0516
6
9773
Sto
reC
om
pu
ters
Pri
nte
rs
A12
810
1
B20
516
6
C97
73
un
def
ined
3 �
4
1 �
14
�4
un
def
ined
un
def
ined
2 �
23
�3
©G
lenc
oe/M
cGra
w-H
ill18
6G
lenc
oe A
lgeb
ra 2
Fo
urt
h-O
rder
Det
erm
inan
tsT
o fi
nd
the
valu
e of
a 4
�4
dete
rmin
ant,
use
a m
eth
od c
alle
d ex
pan
sion
by
min
ors.
Fir
st w
rite
th
e ex
pan
sion
.Use
th
e fi
rst
row
of
the
dete
rmin
ant.
Rem
embe
r th
at t
he
sign
s of
th
e te
rms
alte
rnat
e.
���
��
��
��
�T
hen
eva
luat
e ea
ch 3
�3
dete
rmin
ant.
Use
an
y ro
w.
��
��
��
��
��
�2(
�16
�10
) �
�3(
24)
�5(
�6)
��
52�
�10
2
��
��
��
��
��
�6(
�16
�10
)�
�4(
�6)
�3(
�12
)�
�15
6�
�12
Fin
ally
,eva
luat
e th
e or
igin
al 4
�4
dete
rmin
ant.
��
Eva
luat
e ea
ch d
eter
min
ant.
1.
��
2.
��
3.
��
�10
9�
72�
676
14
30
�2
�3
64
51
12
42
5�
1
33
33
21
21
43
�1
52
50
1
12
31
43
�1
02
�5
44
1�
20
2�6(
�52
) �
3(�
102)
�2(
�15
6) �
7(�
12)
��
846
6�
32
70
43
50
21
�4
60
�2
0
02
60
�3
01
6�
2�
�4
04
30
21
60
�2
45
2�
4�
60
45
02
�4
60
0
01
6�
2�
50
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60
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30
35
01
�4
6�
20
45
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2)4
35
21
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0�
20
04
30
21
60
�2
� 7
04
50
2�
46
00
�2
03
50
1�
46
�2
0�
(�
3)4
35
21
�4
0�
20
�6
6�
32
70
43
50
21
�4
60
�2
0
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-4)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Tran
sfo
rmat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
©G
lenc
oe/M
cGra
w-H
ill18
7G
lenc
oe A
lgeb
ra 2
Lesson 4-4
Tran
slat
ion
s an
d D
ilati
on
sM
atri
ces
that
rep
rese
nt
coor
din
ates
of
poin
ts o
n a
pla
ne
are
use
ful
in d
escr
ibin
g tr
ansf
orm
atio
ns.
Tran
slat
ion
a tr
ansf
orm
atio
n th
at m
oves
a f
igur
e fr
om o
ne lo
catio
n to
ano
ther
on
the
coor
dina
te p
lane
You
can
use
mat
rix
addi
tion
an
d a
tran
slat
ion
mat
rix
to f
ind
the
coor
din
ates
of
the
tran
slat
ed f
igu
re.
Dila
tio
na
tran
sfor
mat
ion
in w
hich
a f
igur
e is
enl
arge
d or
red
uced
You
can
use
sca
lar
mu
ltip
lica
tion
to
perf
orm
dil
atio
ns.
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
�A
BC
wit
h v
erti
ces
A(�
5,4)
,B(�
1,5)
,an
d
C(�
3,�
1) i
f it
is
mov
ed 6
un
its
to t
he
righ
t an
d 4
un
its
dow
n.T
hen
gra
ph
�A
BC
and
its
im
age
�A
�B�C
�.
Wri
te t
he
vert
ex m
atri
x fo
r �
AB
C.
Add
th
e tr
ansl
atio
n m
atri
x to
th
e ve
rtex
m
atri
x of
�A
BC
. �
�
Th
e co
ordi
nat
es o
f th
e ve
rtic
es o
f �
A�B
�C�
are
A�(
1,0)
,B�(
5,1)
,an
d C
�(3,
�5)
.
For
Exe
rcis
es 1
an
d 2
use
th
e fo
llow
ing
info
rmat
ion
.Qu
adri
late
ral
QU
AD
wit
hve
rtic
es Q
(�1,
�3)
,U(0
,0),
A(5
,�1)
,an
d D
(2,�
5) i
s tr
ansl
ated
3 u
nit
s to
th
e le
ftan
d 2
un
its
up
.
1.W
rite
th
e tr
ansl
atio
n m
atri
x.
2.F
ind
the
coor
din
ates
of
the
vert
ices
of
Q�U
�A�D
�.Q
�(�
4,�
1),U
�,(�
3,2)
,A�(
2,1)
,D�(
�1,
�3)
For
Exe
rcis
es 3
–5,u
se t
he
foll
owin
g in
form
atio
n.T
he
vert
ices
of
�A
BC
are
A(4
,�2)
,B(2
,8),
and
C(8
,2).
Th
e tr
ian
gle
is d
ilat
ed s
o th
at i
ts p
erim
eter
is
one-
fou
rth
th
e or
igin
al p
erim
eter
.
3.W
rite
th
e co
ordi
nat
es o
f th
e ve
rtic
es o
f �
AB
Cin
a v
erte
x m
atri
x.
4.F
ind
the
coor
din
ates
of
the
vert
ices
of
imag
e �
A�B
�C�.
A� �1
,��,B
� �,2
�,C� �2
,�
5.G
raph
th
e pr
eim
age
and
the
imag
e.
1 � 21 � 2
1 � 24
28
�2
82
B�
C�
A
B
C
A�
x
y
O
�3
�3
�3
�3
2
22
2
15
3 0
1�
5
66
6 �
4�
4�
4�
5�
1�
3
45
�1
6
66
�4
�4
�4
�5
�1
�3
4
5�
1x
y
OA
�B
�
C�
AB
C
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill18
8G
lenc
oe A
lgeb
ra 2
Ref
lect
ion
s an
d R
ota
tio
ns
Ref
lect
ion
For
a r
efle
ctio
n ov
er t
he:
x-ax
isy-
axis
line
y�
x
Mat
rice
sm
ultip
ly t
he v
erte
x m
atrix
on
the
left
by:
Ro
tati
on
For
a c
ount
ercl
ockw
ise
rota
tion
abou
t th
e90
°18
0°27
0°
Mat
rice
s
orig
in o
f:
mul
tiply
the
ver
tex
mat
rix o
n th
e le
ft by
:
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
�A
BC
wit
h
A(3
,5),
B(�
2,4)
,an
d C
(1,�
1) a
fter
a r
efle
ctio
n o
ver
the
lin
e y
�x.
Wri
te t
he
orde
red
pair
s as
a v
erte
x m
atri
x.T
hen
mu
ltip
ly t
he
vert
ex m
atri
x by
th
ere
flec
tion
mat
rix
for
y�
x.
� �
Th
e co
ordi
nat
es o
f th
e ve
rtic
es o
f A
�B�C
�ar
e A
�(5,
3),B
�(4,
�2)
,an
d C
�(�
1,1)
.
1.T
he
coor
din
ates
of
the
vert
ices
of
quad
rila
tera
l A
BC
Dar
e A
(�2,
1),B
(�1,
3),C
(2,2
),an
dD
(2,�
1).W
hat
are
th
e co
ordi
nat
es o
f th
e ve
rtic
es o
f th
e im
age
A�B
�C�D
�af
ter
are
flec
tion
ove
r th
e y-
axis
?A
�(2,
1),B
�(1,
3),C
�(�
2,2)
,D�(
�2,
�1)
2.T
rian
gle
DE
Fw
ith
ver
tice
s D
(�2,
5),E
(1,4
),an
d F
(0,�
1) i
s ro
tate
d 90
°co
un
terc
lock
wis
e ab
out
the
orig
in.
a.W
rite
th
e co
ordi
nat
es o
f th
e tr
ian
gle
in a
ver
tex
mat
rix.
b.
Wri
te t
he
rota
tion
mat
rix
for
this
sit
uat
ion
.
c.F
ind
the
coor
din
ates
of
the
vert
ices
of
�D
�E�F
�.D
�(�
5,�
2),E
�(�
4,1)
,F�(
1,0)
d.
Gra
ph �
DE
Fan
d �
D�E
�F�.
D�E
�
F�
DE
Fx
y
O
0�
11
0
�2
10
5
4�
1
54
�1
3�
21
3�
21
54
�1
01
10
0
1�
10
�1
0
0�
10
�1
10
01
10
�1
0
01
10
0�
1
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Tran
sfo
rmat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 4-4)
Skil
ls P
ract
ice
Tran
sfo
rmat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
©G
lenc
oe/M
cGra
w-H
ill18
9G
lenc
oe A
lgeb
ra 2
Lesson 4-4
For
Exe
rcis
es 1
–3,u
se t
he
foll
owin
g in
form
atio
n.
Tri
angl
e A
BC
wit
h v
erti
ces
A(2
,3),
B(0
,4),
and
C(�
3,�
3) i
s tr
ansl
ated
3 u
nit
s ri
ght
and
1 u
nit
dow
n.
1.W
rite
th
e tr
ansl
atio
n m
atri
x.
2.F
ind
the
coor
din
ates
of
�A
�B�C
�.A
�(5,
2),B
�(3,
3),C
�(0,
�4)
3.G
raph
th
e pr
eim
age
and
the
imag
e.
For
Exe
rcis
es 4
–6,u
se t
he
foll
owin
g in
form
atio
n.
Th
e ve
rtic
es o
f �
RS
Tar
e R
(�3,
1),S
(2,�
1),a
nd
T(1
,3).
Th
e tr
ian
gle
is d
ilat
ed s
o th
at i
ts p
erim
eter
is
twic
e th
e or
igin
al
peri
met
er.
4.W
rite
th
e co
ordi
nat
es o
f �
RS
Tin
a
vert
ex m
atri
x.
5.F
ind
the
coor
din
ates
of
the
imag
e �
R�S
�T�.
R�(
�6,
2),S
�(4,
�2)
,T�(2
,6)
6.G
raph
�R
ST
and
�R
�S�T
�.
For
Exe
rcis
es 7
–10,
use
th
e fo
llow
ing
info
rmat
ion
.T
he
vert
ices
of
�D
EF
are
D(4
,0),
E(0
,�1)
,an
d F
(2,3
).T
he
tria
ngl
e is
ref
lect
ed o
ver
the
x-ax
is.
7.W
rite
th
e co
ordi
nat
es o
f �
DE
Fin
a
vert
ex m
atri
x.
8.W
rite
th
e re
flec
tion
mat
rix
for
this
sit
uat
ion
.
9.F
ind
the
coor
din
ates
of
�D
�E�F
�.D
�(4,
0),E
�(0,
1),F
�(2,�
3)
10.G
raph
�D
EF
and
�D
�E�F
�.
For
Exe
rcis
es 1
1–14
,use
th
e fo
llow
ing
info
rmat
ion
.T
rian
gle
XY
Zw
ith
ver
tice
s X
(1,�
3),Y
(�4,
1),a
nd
Z(�
2,5)
is
rota
ted
180º
cou
nte
rclo
ckw
ise
abou
t th
e or
igin
.
11.W
rite
th
e co
ordi
nat
es o
f th
e tr
ian
gle
in a
ve
rtex
mat
rix.
12.W
rite
th
e ro
tati
on m
atri
x fo
r th
is s
itu
atio
n.
13.F
ind
the
coor
din
ates
of
�X
�Y�Z
�.X
�(�
1,3)
,Y�(
4,�
1),Z
�(2,
�5)
14.G
raph
th
e pr
eim
age
and
the
imag
e.
�1
0
0�
1
1
�4
�2
�3
15
X�
Y�
Z�
X
Y
Z
x
y
O
10
0�
1D
�E
�
F�
DE
F
x
y
O
40
20
�1
3
�3
21
1
�1
3R
�
S�
T�
R
S
T
x
y
O
3
33
�1
�1
�1
B�
C�
AB
C
A� x
y
O
©G
lenc
oe/M
cGra
w-H
ill19
0G
lenc
oe A
lgeb
ra 2
For
Exe
rcis
es 1
–3,u
se t
he
foll
owin
g in
form
atio
n.
Qu
adri
late
ral
WX
YZ
wit
h v
erti
ces
W(�
3,2)
,X(�
2,4)
,Y(4
,1),
and
Z(3
,0)
is t
ran
slat
ed 1
un
it l
eft
and
3 u
nit
s do
wn
.
1.W
rite
th
e tr
ansl
atio
n m
atri
x.
2.F
ind
the
coor
din
ates
of
quad
rila
tera
l W
�X�Y
�Z�.
W�(�
4,�
1),X
�(�3,
1),Y
�(3,�
2),Z
�(2,�
3)3.
Gra
ph t
he
prei
mag
e an
d th
e im
age.
For
Exe
rcis
es 4
–6,u
se t
he
foll
owin
g in
form
atio
n.
Th
e ve
rtic
es o
f �
RS
Tar
e R
(6,2
),S
(3,�
3),a
nd
T(�
2,5)
.Th
e tr
ian
gle
is d
ilat
ed s
o th
at i
ts p
erim
eter
is
one
hal
f th
e or
igin
al p
erim
eter
.
4.W
rite
th
e co
ordi
nat
es o
f �
RS
Tin
a v
erte
x m
atri
x.
5.F
ind
the
coor
din
ates
of
the
imag
e �
R�S
�T�.
R�(3
,1),
S�(1
.5,�
1.5)
,T�(�
1,2.
5)6.
Gra
ph �
RS
Tan
d �
R�S
�T�.
For
Exe
rcis
es 7
–10,
use
th
e fo
llow
ing
info
rmat
ion
.T
he
vert
ices
of
quad
rila
tera
l A
BC
Dar
e A
(�3,
2),B
(0,3
),C
(4,�
4),
and
D(�
2,�
2).T
he
quad
rila
tera
l is
ref
lect
ed o
ver
the
y-ax
is.
7.W
rite
th
e co
ordi
nat
es o
f A
BC
Din
a
vert
ex m
atri
x.
8.W
rite
th
e re
flec
tion
mat
rix
for
this
sit
uat
ion
.
9.F
ind
the
coor
din
ates
of
A�B
�C�D
�.A
�(3,2
),B
�(0,3
),C
�(�4,
�4)
,D�(2
,�2)
10.G
raph
AB
CD
and
A�B
�C�D
�.
11.A
RC
HIT
ECTU
RE
Usi
ng
arch
itec
tura
l de
sign
sof
twar
e,th
e B
radl
eys
plot
th
eir
kitc
hen
plan
s on
a g
rid
wit
h e
ach
un
it r
epre
sen
tin
g 1
foot
.Th
ey p
lace
th
e co
rner
s of
an
isl
and
at(2
,8),
(8,1
1),(
3,5)
,an
d (9
,8).
If t
he
Bra
dley
s w
ish
to
mov
e th
e is
lan
d 1.
5 fe
et t
o th
eri
ght
and
2 fe
et d
own
,wh
at w
ill
the
new
coo
rdin
ates
of
its
corn
ers
be?
(3.5
,6),
(9.5
,9),
(4.5
,3),
and
(10
.5,6
)
12.B
USI
NES
ST
he
desi
gn o
f a
busi
nes
s lo
go c
alls
for
loc
atin
g th
e ve
rtic
es o
f a
tria
ngl
e at
(1.5
,5),
(4,1
),an
d (1
,0)
on a
gri
d.If
des
ign
ch
ange
s re
quir
e ro
tati
ng
the
tria
ngl
e 90
ºco
un
terc
lock
wis
e,w
hat
wil
l th
e n
ew c
oord
inat
es o
f th
e ve
rtic
es b
e?(�
5,1.
5),(
�1,
4),a
nd
(0,
1)
�1
0
01
A�
B�
C�
D�
AB
C
D
x
y
O
�3
04
�2
2
3�
4�
2
63
�2
2�
35
R�
S�
T�
R
S
T
x
y
O
�1
�1
�1
�1
�3
�3
�3
�3
W�X
�
Y�
Z�
W
X
Y
Zx
y
O
Pra
ctic
e (
Ave
rag
e)
Tran
sfo
rmat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-4)
Readin
g t
o L
earn
Math
em
ati
csTr
ansf
orm
atio
ns
wit
h M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
©G
lenc
oe/M
cGra
w-H
ill19
1G
lenc
oe A
lgeb
ra 2
Lesson 4-4
Pre-
Act
ivit
yH
ow a
re t
ran
sfor
mat
ion
s u
sed
in
com
pu
ter
anim
atio
n?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-4
at
the
top
of p
age
175
in y
our
text
book
.
Des
crib
e h
ow y
ou c
an c
han
ge t
he
orie
nta
tion
of
a fi
gure
wit
hou
t ch
angi
ng
its
size
or
shap
e.
Flip
(o
r re
flec
t) t
he
fig
ure
ove
r a
line.
Rea
din
g t
he
Less
on
1.a.
Wri
te t
he
vert
ex m
atri
x fo
r th
e qu
adri
late
ral
AB
CD
show
n i
n
the
grap
h a
t th
e ri
ght.
b.
Wri
te t
he v
erte
x m
atri
x th
at r
epre
sent
s th
e po
siti
on o
f th
e qu
adri
late
ral
A�B
�C�D
�th
atre
sult
s w
hen
qu
adri
late
ral
AB
CD
is t
ran
slat
ed 3
un
its
to t
he
righ
t an
d 2
un
its
dow
n.
2.D
escr
ibe
the
tran
sfor
mat
ion
th
at c
orre
spon
ds t
o ea
ch o
f th
e fo
llow
ing
mat
rice
s.
a.b
.
cou
nte
rclo
ckw
ise
rota
tio
n
tran
slat
ion
4 u
nit
s d
ow
n a
nd
ab
ou
t th
e o
rig
in o
f 18
0�3
un
its
to t
he
rig
ht
c.d
.
refl
ecti
on
ove
r th
e y-
axis
refl
ecti
on
ove
r th
e lin
e y
�x
Hel
pin
g Y
ou
Rem
emb
er
3.D
escr
ibe
a w
ay t
o re
mem
ber
wh
ich
of
the
refl
ecti
on m
atri
ces
corr
espo
nds
to
refl
ecti
onov
er t
he
x-ax
is.
Sam
ple
an
swer
:Th
e o
nly
ele
men
ts u
sed
in t
he
refl
ecti
on
mat
rice
s ar
e 0,
1,an
d �
1.F
or
such
a 2
�2
mat
rix
Mto
hav
e th
e p
rop
erty
th
at
M�
�,t
he
elem
ents
in t
he
top
ro
w m
ust
be
1 an
d 0
(in
th
at
ord
er),
and
ele
men
ts in
th
e b
ott
om
ro
w m
ust
be
0 an
d �
1 (i
n t
hat
ord
er).
x
�y
x
y
01
10
�1
0
01
3
33
�4
�4
�4
�1
0
0�
1
�1
54
1�
11
�6
�5
�4
21
�2
1
3�
4�
3x
y
O
A
B
CD
©G
lenc
oe/M
cGra
w-H
ill19
2G
lenc
oe A
lgeb
ra 2
Pro
per
ties
of
Det
erm
inan
tsT
he fo
llow
ing
prop
erti
es o
ften
hel
p w
hen
eval
uati
ng d
eter
min
ants
.
•If
all
th
e el
emen
ts o
f a
row
(or
col
um
n)
are
zero
,th
e va
lue
of t
he
dete
rmin
ant
is z
ero.
�0
(a�
0) �
(0 �
b)
�0
•M
ult
iply
ing
all
the
elem
ents
of
a ro
w (
or c
olu
mn
) by
a c
onst
ant
is e
quiv
alen
t to
mu
ltip
lyin
g th
e va
lue
of t
he
dete
rmin
ant
by t
he
con
stan
t.
3�
3[4(
3) �
5(�
1)]
�51
3[12
�5]
�51
12(3
) �
5(�
3)�
51
•If
tw
o ro
ws
(or
colu
mn
s) h
ave
equ
al c
orre
spon
din
g el
emen
ts,t
he
valu
e of
th
ede
term
inan
t is
zer
o.
�0
5(�
3) �
(�3)
(5)
�0
•T
he
valu
e of
a d
eter
min
ant
is u
nch
ange
d if
an
y m
ult
iple
of
a ro
w (
or c
olu
mn
) is
add
ed t
o co
rres
pon
din
g el
emen
ts o
f an
oth
er r
ow (
or c
olu
mn
).
�4(
5) �
2(�
3)�
6(5)
�2(
2)�
2620
�6
�26
30 �
4�
26(R
ow 2
is a
dded
to
row
1.)
•If
tw
o ro
ws
(or
colu
mn
s) a
re i
nte
rch
ange
d,th
e si
gn o
f th
e de
term
inan
t is
ch
ange
d.
�4(
8) �
(�3)
(5)
��
[(�
3)(5
) �
4(8)
]�
4732
�15
�47
�[�
15 �
32]
�47
•T
he
valu
e of
th
e de
term
inan
t is
un
chan
ged
if r
ow 1
is
inte
rch
ange
d w
ith
col
um
n 1
,an
d ro
w 2
is
inte
rch
ange
d w
ith
col
um
n 2
.Th
e re
sult
is
call
ed t
he
tran
spos
e.
�5(
4) �
3(�
7)�
5(4)
�(�
7)(3
)�
20 �
21�
4120
�21
�41
Exe
rcis
es 1
–6
Ver
ify
each
pro
pert
y ab
ove
by e
valu
atin
g th
e gi
ven
det
erm
inan
ts a
nd
give
an
oth
er e
xam
ple
of t
he
prop
erty
.E
xam
ple
s w
ill v
ary.
5
3 �
74
5�
7 3
4
�3
8
55
4
5 �
38
62
25
4�
3 2
5
5
5 �
3�
3
12
�3
5
34
�1
53
ab
00
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 4-5)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Det
erm
inan
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
©G
lenc
oe/M
cGra
w-H
ill19
3G
lenc
oe A
lgeb
ra 2
Lesson 4-5
Det
erm
inan
ts o
f 2
�2
Mat
rice
s
Sec
on
d-O
rder
Det
erm
inan
tF
or t
he m
atrix
,
the
dete
rmin
ant
is
�ad
�bc
.
Fin
d t
he
valu
e of
eac
h d
eter
min
ant.
a.
�6(
5) �
3(�
8)
�30
�(�
24)
or 5
4
b.
�11
(�3)
�(�
5)(9
)
��
33 �
(�45
) or
12
Fin
d t
he
valu
e of
eac
h d
eter
min
ant.
1.52
2.�
23.
�18
4.64
5.�
66.
1
7.�
114
8.14
49.
�33
5
10.
4811
.�
1612
.49
5
13.
1514
.36
15.
17
16.
18.3
17.
18.
37
6.8
15
�0.
25
13 � 60�2 3�
��1 2�
�1 6��1 5�
4.8
2.1
3.4
5.3
20
110
0.1
1.4
8.6
0.5
14
5
0.2
8 �
1.5
15
13
62
�4
19
24
�8
7
�3
10
16
22
40
�8
35
5
20
15
12
23
28
14
8
9�
3
47
59
�6
�3
�4
�1
5
12
�7
�4
39
46
�8
3 �
21
6�
2 5
7
11
�5
9
3
11
�5
9
3
6
3 �
85
6
3�
85
ab
c
d
ab
c
d
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill19
4G
lenc
oe A
lgeb
ra 2
Det
erm
inan
ts o
f 3
�3
Mat
rice
s
Th
ird
-Ord
er D
eter
min
ants
�a
�b
�c
Are
a o
f a
Tria
ng
le
The
are
a of
a t
riang
le h
avin
g ve
rtic
es (
a, b
), (
c, d
) an
d (e
, f)
is | A
| , w
here
A�
.
Eva
luat
e .
�4
�5
�2
Thi
rd-o
rder
det
erm
inan
t
�4(
18 �
0) �
5(6
�0)
�2(
�3
�6)
Eva
luat
e 2
�2
dete
rmin
ants
.
�4(
18)
�5(
6) �
2(�
9)S
impl
ify.
�72
�30
�18
Mul
tiply
.
�60
Sim
plify
.
Eva
luat
e ea
ch d
eter
min
ant.
1.�
572.
803.
28
4.54
5.�
636.
�2
7.F
ind
the
area
of
a tr
ian
gle
wit
h v
erti
ces
X(2
,�3)
,Y(7
,4),
and
Z(�
5,5)
.44
.5 s
qu
are
un
its
5
�4
1
23
�2
�1
6�
3
6
1�
4
32
1 �
22
�1
5�
22
30
�2
24
�3
6
14
�2
30
�1
32
4
10
�2
31
2
�2
5
3
�2
�2
0
41
�1
5�
3
13
2�
31
0 2
6
30
�3
64
5�
2 1
30
2�
36
45
�2
13
0 2
�3
6
ab
1
cd
1
ef
1
1 � 2
de
g
h
df
gi
ef
hi
ab
c
de
f
gh
i
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Det
erm
inan
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-5)
Skil
ls P
ract
ice
Det
erm
inan
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
©G
lenc
oe/M
cGra
w-H
ill19
5G
lenc
oe A
lgeb
ra 2
Lesson 4-5
Fin
d t
he
valu
e of
eac
h d
eter
min
ant.
1.13
2.35
3.1
4.�
135.
�45
6.0
7.14
8.13
9.1
10.
1111
.�
512
.�
52
13.
�3
14.
1715
.68
16.
�4
17.
1018
.�
17
Eva
luat
e ea
ch d
eter
min
ant
usi
ng
exp
ansi
on b
y m
inor
s.
19.
�1
20.
�2
21.
�1
Eva
luat
e ea
ch d
eter
min
ant
usi
ng
dia
gon
als.
22.
�3
23.
824
.40
32
2 1
�1
4 3
�1
0
3�
12
10
4 3
�2
0
2�
16
32
5 2
31
26
1 3
5�
1 2
1�
2
6�
11
52
�1
13
�2
2�
11
32
�1
23
�2
�1
6
25
2
2 �
14
�1
2
04
�1
�14
52
�1
�3
5
�2
3�
5 6
�11
�12
4
14
1
�3
�3
41
�5
16
9
�2
�4
1�
31
8
�7
�5
2
8�
6
312
2
80
9 5
82
5 3
1
16
17
10
9
58
52
13
©G
lenc
oe/M
cGra
w-H
ill19
6G
lenc
oe A
lgeb
ra 2
Fin
d t
he
valu
e of
eac
h d
eter
min
ant.
1.�
52.
03.
�18
4.34
5.�
206.
3
7.36
8.55
9.11
2
10.
3011
.�
1612
.0.
13
Eva
luat
e ea
ch d
eter
min
ant
usi
ng
exp
ansi
on b
y m
inor
s.
13.
�48
14.
4515
.7
16.
2817
.�
7218
.31
8
Eva
luat
e ea
ch d
eter
min
ant
usi
ng
dia
gon
als.
19.
1020
.12
21.
5
22.
2023
.�
424
.0
25.G
EOM
ETRY
Fin
d th
e ar
ea o
f a
tria
ngl
e w
hos
e ve
rtic
es h
ave
coor
din
ates
(3,
5),(
6,�
5),
and
(�4,
10).
27.5
un
its2
26.L
AN
D M
AN
AG
EMEN
TA
fis
h a
nd
wil
dlif
e m
anag
emen
t or
gan
izat
ion
use
s a
GIS
(geo
grap
hic
in
form
atio
n s
yste
m)
to s
tore
an
d an
alyz
e da
ta f
or t
he
parc
els
of l
and
itm
anag
es.A
ll o
f th
e pa
rcel
s ar
e m
appe
d on
a g
rid
in w
hic
h 1
un
it r
epre
sen
ts 1
acr
e.If
the
coor
din
ates
of
the
corn
ers
of a
par
cel
are
(�8,
10),
(6,1
7),a
nd
(2,�
4),h
ow m
any
acre
s is
th
e pa
rcel
?13
3 ac
res
21
3 1
80
05
�1
2�
1�
2 4
0�
2 0
32
12
�4
14
�6
23
3
1�
4�
1 1
�6
�2
23
1
22
3 1
�1
1 3
11
�4
3�
1
21
�2
4
1�
4
�12
03
7
5�
1
42
�6
27
�6
84
0 1
�1
3
0�
40
2�
11
3�
25
21
1 1
�1
�2
11
�1
2
�4
1
30
9 �
15
7
�2
31
0
4�
3
25
�1
0.5
�0.
7 0
.4�
0.3
2�
1 3
�9.
5
3�
4 3
.75
5
�1
�11
1
0�
23
�4
79
4
0 �
29
2�
5 5
�11
4�
3 �
124
�14
�3
2
�2
4
1 �
2�
59
6 3
21
6 2
7
Pra
ctic
e (
Ave
rag
e)
Det
erm
inan
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 4-5)
Readin
g t
o L
earn
Math
em
ati
csD
eter
min
ants
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
©G
lenc
oe/M
cGra
w-H
ill19
7G
lenc
oe A
lgeb
ra 2
Lesson 4-5
Pre-
Act
ivit
yH
ow a
re d
eter
min
ants
use
d t
o fi
nd
are
as o
f p
olyg
ons?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-5
at
the
top
of p
age
182
in y
our
text
book
.
In t
his
less
on,y
ou w
ill le
arn
how
to
find
the
are
a of
a t
rian
gle
if y
ou k
now
the
coor
dina
tes
of it
s ve
rtic
es u
sing
det
erm
inan
ts.D
escr
ibe
a m
etho
d yo
u al
read
ykn
ow f
or f
indi
ng
the
area
of
the
Ber
mu
da T
rian
gle.
Sam
ple
an
swer
:U
se t
he
map
.Ch
oo
se a
ny s
ide
of
the
tria
ng
le a
s th
e b
ase,
and
mea
sure
th
is s
ide
wit
h a
ru
ler.
Mu
ltip
ly t
his
len
gth
by
the
scal
efa
cto
r fo
r th
e m
ap.N
ext,
dra
w a
seg
men
t fr
om
th
e o
pp
osi
teve
rtex
per
pen
dic
ula
r to
th
e b
ase.
Mea
sure
th
is s
egm
ent,
and
mu
ltip
ly it
s le
ng
th b
y th
e sc
ale
for
the
map
.Fin
ally
,fin
d t
he
area
by
usi
ng
th
e fo
rmu
la A
�b
h.
Rea
din
g t
he
Less
on
1.In
dica
te w
het
her
eac
h o
f th
e fo
llow
ing
stat
emen
ts i
s tr
ue
or f
alse
.
a.E
very
mat
rix
has
a d
eter
min
ant.
fals
e
b.
If b
oth
row
s of
a 2
�2
mat
rix
are
iden
tica
l,th
e de
term
inan
t of
the
mat
rix
will
be
0.tr
ue
c.E
very
ele
men
t of
a 3
�3
mat
rix
has
a m
inor
.tr
ue
d.I
n o
rder
to
eval
uat
e a
thir
d-or
der
dete
rmin
ant
by e
xpan
sion
by
min
ors
it i
s n
eces
sary
to f
ind
the
min
or o
f ev
ery
elem
ent
of t
he
mat
rix.
fals
e
e.If
you
eva
luat
e a
thir
d-or
der
dete
rmin
ant
by e
xpan
sion
abo
ut
the
seco
nd
row
,th
epo
siti
on s
ign
s yo
u w
ill
use
are
��
�.
tru
e
2.S
upp
ose
that
tri
angl
e R
ST
has
ver
tice
s R
(�2,
5),S
(4,1
),an
d T
(0,6
).
a.W
rite
a d
eter
min
ant
that
cou
ld b
e u
sed
in f
indi
ng
the
area
of
tria
ngl
e R
ST
.
b.
Exp
lain
how
you
wou
ld u
se t
he
dete
rmin
ant
you
wro
te i
n p
art
ato
fin
d th
e ar
ea o
fth
e tr
ian
gle.
Sam
ple
an
swer
:E
valu
ate
the
det
erm
inan
t an
d m
ult
iply
th
e re
sult
by
.Th
en t
ake
the
abso
lute
val
ue
to m
ake
sure
th
e fi
nal
an
swer
is p
osi
tive
.
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
a co
mpl
icat
ed p
roce
dure
is
to b
reak
it
dow
n i
nto
ste
ps.W
rite
ali
st o
f st
eps
for
eval
uat
ing
a th
ird-
orde
r de
term
inan
t u
sin
g ex
pan
sion
by
min
ors.
Sam
ple
an
swer
:1.
Ch
oo
se a
ro
w o
f th
e m
atri
x.2.
Fin
d t
he
po
siti
on
sig
ns
for
the
row
yo
u h
ave
cho
sen
.3.F
ind
th
e m
ino
r o
f ea
ch e
lem
ent
in t
hat
row
.4.M
ult
iply
eac
h e
lem
ent
by it
s p
osi
tio
n s
ign
an
d b
y it
s m
ino
r.5.
Ad
dth
e re
sult
s.
1 � 2
�2
51
4
11
0
61
1 � 2
©G
lenc
oe/M
cGra
w-H
ill19
8G
lenc
oe A
lgeb
ra 2
Mat
rix
Mu
ltip
licat
ion
A f
urn
itu
re m
anu
fact
ure
r m
akes
uph
olst
ered
ch
airs
an
d w
ood
tabl
es.M
atri
x A
show
s th
e n
um
ber
of h
ours
spe
nt
on e
ach
ite
m b
y th
ree
diff
eren
t w
orke
rs.O
ne
day
the
fact
ory
rece
ives
an
ord
er f
or 1
0 ch
airs
an
d 3
tabl
es.T
his
is
show
n i
n
mat
rix
B.
�A
�B
[10
3]�
[10(
4) �
3(18
)10
(2)
�3(
15)
10(1
2) �
3(0)
] �
[94
6512
0]
Th
e pr
odu
ct o
f th
e tw
o m
atri
ces
show
s th
e n
um
ber
of h
ours
nee
ded
for
each
ty
pe o
f w
orke
r to
com
plet
e th
e or
der:
94 h
ours
for
woo
dwor
king
,65
hour
s fo
r fi
nish
ing,
and
120
hour
s fo
r up
hols
teri
ng.
To
fin
d th
e to
tal
labo
r co
st,m
ult
iply
by
a m
atri
x th
at s
how
s th
e h
ourl
y ra
te f
or
each
wor
ker:
$15
for
woo
dwor
kin
g,$9
for
fin
ish
ing,
and
$12
for
uph
olst
erin
g.
C�
[94
6512
0]
�[9
4(15
) �
65(9
) �
120(
12)]
�$3
435
Use
mat
rix
mu
ltip
lica
tion
to
solv
e th
ese
pro
ble
ms.
A c
andy
com
pan
y pa
ckag
es c
aram
els,
choc
olat
es,a
nd
har
d ca
ndy
in
th
ree
diff
eren
t as
sort
men
ts:t
radi
tion
al,d
elu
xe,a
nd
supe
rb.F
or e
ach
typ
e of
can
dy
the
tabl
e be
low
giv
es t
he
nu
mbe
r in
eac
h a
ssor
tmen
t,th
e n
um
ber
of C
alor
ies
per
piec
e,an
d th
e co
st t
o m
ake
each
pie
ce.
Cal
orie
sco
st p
er
trad
itio
nal
delu
xesu
perb
per
piec
epi
ece
(cen
ts)
cara
mel
s10
1615
6010
choc
olat
es12
825
7012
card
can
dy10
168
556
Th
e co
mpa
ny
rece
ives
an
ord
er f
or 3
00 t
radi
tion
al,1
80 d
elu
xe a
nd
100
supe
rb a
ssor
tmen
ts.
1.F
ind
the
nu
mbe
r of
eac
h t
ype
of c
andy
nee
ded
to f
ill
the
orde
r.73
80 c
aram
els;
7540
ch
oco
late
s;66
80 h
ard
can
die
s
2.F
ind
the
tota
l n
um
ber
of C
alor
ies
in e
ach
typ
e of
ass
ortm
ent.
1990
-tra
dit
ion
al;
2400
-del
uxe
;30
90-s
up
erb
3.F
ind
the
cost
of
prod
uct
ion
for
eac
h t
ype
of a
ssor
tmen
t.$3
.04-
trad
itio
nal
;$3
.52-
del
uxe
;$4
.98-
sup
erb
4.F
ind
the
cost
to
fill
th
e or
der.
$204
3.60
15
9
12
42
12
18
150
chai
rta
ble
num
ber
orde
red
[10
3]
hour
sw
oodw
orke
rfin
sher
upho
lste
rer
chai
r
42
12
tabl
e
1815
0
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-6)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Cra
mer
’s R
ule
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
©G
lenc
oe/M
cGra
w-H
ill19
9G
lenc
oe A
lgeb
ra 2
Lesson 4-6
Syst
ems
of
Two
Lin
ear
Equ
atio
ns
Det
erm
inan
ts p
rovi
de a
way
for
sol
vin
g sy
stem
sof
equ
atio
ns.
Cra
mer
’s R
ule
fo
r
The
sol
utio
n of
the
line
ar s
yste
m o
f eq
uatio
nsax
�by
�e
Two
-Var
iab
le S
yste
ms
cx�
dy�
f
is (
x, y
) w
here
x�
,
y�
, an
d �
0.
Use
Cra
mer
’s R
ule
to
solv
e th
e sy
stem
of
equ
atio
ns.
5x�
10y
�8
10x
�25
y�
�2
x�
Cra
mer
’s R
ule
y�
�
a�
5, b
��
10,
c�
10,
d�
25,
e�
8,
f�
�2
�
�
Eva
luat
e ea
ch d
eter
min
ant.
�
�or
S
impl
ify.
��
or �
Th
e so
luti
on i
s �
,��.
Use
Cra
mer
’s R
ule
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.
1.3x
�2y
�7
2.x
�4y
�17
3.2x
�y
��
22x
�7y
�38
(5,4
)3x
�y
�29
(9,�
2)4x
�y
�4
(3,8
)
4.2x
�y
�1
5.4x
�2y
�1
6.6x
�3y
��
35x
�2y
��
29(�
3,�
7)5x
�4y
�24
�2,�
�2x
�y
�21
(5,1
1)
7.2x
�7y
�16
8.2x
�3y
��
29.
��
2
x�
2y�
30(2
2,�
4)3x
�4y
�9
(35,
24)
��
�8
(�12
,30)
10.6
x�
9y�
�1
11.3
x�
12y
��
1412
.8x
�2y
�
3x�
18y
�12
9x�
6y�
�7
5x�
4y�
�
�,
���
,�
��,
�11 � 14
1 � 75 � 6
4 � 35 � 9
2 � 3
27 � 7
3 � 7
y � 6x � 4
y � 5x � 3
7 � 2
2 � 54 � 5
2 � 590 � 22
54 � 5
180
� 225
5(�
2)�
8(10
)�
��
5(25
)�
(�10
)(10
)8(
25)
�(�
2)(�
10)
��
�5(
25)
�(�
10)(
10)
5
8 1
0�
2�
�
5�
10
10
25
8
�10
�
225
�
�
5�
10
10
25
ae
c
f � a
b
cd
eb
f
d
� ab
c
d
ab
c
d
ae
c
f � a
b
cd
eb
f
d
� ab
c
d
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill20
0G
lenc
oe A
lgeb
ra 2
Syst
ems
of
Thre
e Li
nea
r Eq
uat
ion
s
The
sol
utio
n of
the
sys
tem
who
se e
quat
ions
are
ax�
by�
cz�
j
Cra
mer
’s R
ule
fo
r
dx�
ey�
fz�
k
Th
ree-
Var
iab
le S
yste
ms
gx�
hy�
iz�
l
is (
x, y
, z)
whe
re x
�,
y�
, an
d z
�
and
�0.
Use
Cra
mer
’s r
ule
to
solv
e th
e sy
stem
of
equ
atio
ns.
6x�
4y�
z�
52x
�3y
�2z
��
28x
�2y
�2z
�10
Use
th
e co
effi
cien
ts a
nd
con
stan
ts f
rom
th
e eq
uat
ion
s to
for
m t
he
dete
rmin
ants
.Th
enev
alu
ate
each
det
erm
inan
t.
x�
y
�z
�
�or
�
or �
�or
Th
e so
luti
on i
s �
,�,
�.
Use
Cra
mer
’s r
ule
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.
1.x
�2y
�3z
�6
2.3x
�y
�2z
��
22x
�y
�z
��
34x
�2y
�5z
�7
x�
y�
z�
6(1
,2,3
)x
�y
�z
�1
(3,�
5,3)
3.x
�3y
�z
�1
4.2x
�y
�3z
��
52x
�2y
�z
��
8x
�y
�5z
�21
4x�
7y�
2z�
11(�
2,1,
6)3x
�2y
�4z
�6
(4,7
,�2)
5.3x
�y
�4z
�7
6.2x
�y
�4z
�9
2x�
y�
5z�
�24
3x�
2y�
5z�
�13
10x
�3y
�2z
��
2(�
3,8,
�2)
x�
y�
7z�
0(5
,9,2
)
4 � 31 � 3
5 � 6
4 � 3�
128
� �96
1 � 332� �
965 � 6
�80
� �96
64
5 2
3�
2 8
�2
10
��
64
1 2
3�
2 8
�2
2
65
1 2
�2
�2
810
2�
� 6
41
23
�2
8�
22
5
41
�2
3�
2 1
0�
22
��
6
41
2
3�
2
8�
22
ab
c
de
f
gh
i
ab
j d
ek
g
hl
� ab
c
de
f g
hi
aj
c
dk
f g
li
� ab
c
de
f g
hi
jb
c
ke
f l
hi
� ab
c
de
f g
hi
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Cra
mer
’s R
ule
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
Answers (Lesson 4-6)
Skil
ls P
ract
ice
Cra
mer
’s R
ule
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
©G
lenc
oe/M
cGra
w-H
ill20
1G
lenc
oe A
lgeb
ra 2
Lesson 4-6
Use
Cra
mer
’s R
ule
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.
1.2a
�3b
�6
2.3x
�y
�2
2a�
b�
�2
(�3,
4)2x
�y
�3
(1,�
1)
3.2m
�3n
��
64.
x�
y�
2m
�3n
�6
(0,�
2)2x
�3y
�9
(3,1
)
5.2x
�y
�4
6.3r
�s
�7
7x�
2y�
3(1
,2)
5r�
2s�
8(6
,11)
7.4g
�5h
�1
8.7x
�5y
��
8g
�3h
�2
(�1,
1)9x
�2y
�3
(1,�
3)
9.3x
�4y
�2
10.2
x�
y�
54x
�3y
�12
(6,4
)3x
�y
�5
(2,�
1)
11.3
p�
6q�
1812
.x�
2y�
�1
2p�
3q�
5(4
,�1)
2x�
y�
3(1
,1)
13.5
c�
3d�
514
.5t
�2v
�2
2c�
9d�
2(1
,0)
2t�
3v�
�8
(2,�
4)
15.5
a�
2b�
1416
.65w
�8z
�83
3a�
4b�
11(3
,0.5
)9w
�4z
�0
(1,�
2.25
)
17.G
EOM
ETRY
Th
e tw
o si
des
of a
n a
ngl
e ar
e co
nta
ined
in
th
e li
nes
wh
ose
equ
atio
ns
are
3x�
2y�
4 an
d x
�3y
�5.
Fin
d th
e co
ordi
nat
es o
f th
e ve
rtex
of
the
angl
e.(2
,�1)
Use
Cra
mer
’s R
ule
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.
18.a
�b
�5c
�2
19.x
�3y
�z
�5
3a�
b�
2c�
32x
�5y
�z
�12
4a�
2b�
c�
�3
(2,�
5,1)
x�
2y�
3z�
�13
( 3,
2,4)
20.3
c�
5d
�2e
�4
21.r
�4s
�t
�6
2c�
3d�
4c�
�3
2r�
s�
3t�
04c
�2d
�3e
�0
(1,�
1,�
2)3r
�2s
�t
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(1,�
1,�
1)
©G
lenc
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cGra
w-H
ill20
2G
lenc
oe A
lgeb
ra 2
Use
Cra
mer
’s R
ule
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.
1.2x
�y
�0
2.5c
�9d
�19
3.2x
�3y
�5
3x�
2y�
�2
(2,�
4)2c
�d
��
20(�
7,6)
3x�
2y�
1(1
,1)
4.20
m�
3n�
285.
x�
3y�
66.
5x�
6y�
�45
2m�
3n�
16(2
,4)
3x�
y�
�22
(�6,
�4)
9x�
8y�
13(�
3,5)
7.�
2e�
f�
48.
2x�
y�
�1
9.8a
�3b
�24
�3e
�5f
��
15(�
5,�
6)2x
�4y
�8
(�2,
�3)
2a�
b�
4(6
,�8)
10.�
3x�
15y
�45
11.3
u�
5v�
1112
.�6g
�h
��
10�
2x�
7y�
18(5
,4)
6u�
7v�
�12
�,�
2 ��
3g�
4h�
4�
,�2 �
13.x
�3y
�8
14.0
.2x
�0.
5y�
�1
15.0
.3d
�0.
6g�
1.8
x�
0.5y
�3
(2,�
2)0.
6x�
3y�
�9
(5,4
)0.
2d�
0.3g
�0.
5(4
,�1)
16.G
EOM
ETRY
Th
e tw
o si
des
of a
n a
ngl
e ar
e co
nta
ined
in
th
e li
nes
wh
ose
equ
atio
ns
are
x�
y�
6 an
d 2x
�y
�1.
Fin
d th
e co
ordi
nat
es o
f th
e ve
rtex
of
the
angl
e.(2
,�3)
17.G
EOM
ETRY
Tw
o si
des
of a
par
alle
logr
am a
re c
onta
ined
in
th
e li
nes
wh
ose
equ
atio
ns
are
0.2x
�0.
5y�
1 an
d 0.
02x
�0.
3y�
�0.
9.F
ind
the
coor
din
ates
of
a ve
rtex
of
the
para
llel
ogra
m.
(15,
4)
Use
Cra
mer
’s R
ule
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.
18.x
�3y
�3z
�4
19.�
5a�
b�
4c�
720
.2x
�y
�3z
��
5�
x�
2y�
z�
�1
�3a
�2b
�c
�0
5x�
2y�
2z�
84x
�y
�2z
��
12a
�3b
�c
�17
3x�
3y�
5z�
17
(1,�
1,2)
(3,2
,�5)
(2,3
,4)
21.2
c�
3d�
e�
1722
.2j
�k
�3m
��
323
.3x
�2y
�5z
�3
4c�
d�
5e�
�9
3j�
2k�
4m�
52x
�2y
�4z
�3
c�
2d�
e�
12�
4j�
k�
2m�
4�
5x�
10y
�7z
��
3
(2,3
,�4)
(�1,
2,1)
(1.2
,0.3
,0)
24.L
AN
DSC
API
NG
A m
emor
ial
gard
en b
ein
g pl
ante
d in
fr
ont
of a
mu
nic
ipal
lib
rary
wil
l co
nta
in t
hre
e ci
rcu
lar
beds
th
at a
re t
ange
nt
to e
ach
oth
er.A
lan
dsca
pe a
rch
itec
t h
as p
repa
red
a sk
etch
of
the
desi
gn f
or t
he
gard
en u
sin
g C
AD
(co
mpu
ter-
aide
d dr
afti
ng)
sof
twar
e,as
sh
own
at
the
righ
t.T
he
cen
ters
of
the
thre
e ci
rcu
lar
beds
are
rep
rese
nte
dby
poi
nts
A,B
,an
d C
.Th
e di
stan
ce f
rom
Ato
Bis
15
feet
,th
e di
stan
ce f
rom
Bto
Cis
13
feet
,an
d th
e di
stan
ce f
rom
A
to C
is 1
6 fe
et.W
hat
is
the
radi
us
of e
ach
of
the
circ
ula
rbe
ds?
circ
le A
:9
ft,c
ircl
e B
:6
ft,c
ircl
e C
:7
ft
A
BC
13 ft
16 ft
15 ft
4 � 3
4 � 31 � 3
Pra
ctic
e (
Ave
rag
e)
Cra
mer
’s R
ule
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-6)
Readin
g t
o L
earn
Math
em
ati
csC
ram
er’s
Ru
le
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
©G
lenc
oe/M
cGra
w-H
ill20
3G
lenc
oe A
lgeb
ra 2
Lesson 4-6
Pre-
Act
ivit
yH
ow i
s C
ram
er’s
Ru
le u
sed
to
solv
e sy
stem
s of
eq
uat
ion
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-6
at
the
top
of p
age
189
in y
our
text
book
.
A t
rian
gle
is b
oun
ded
by t
he
x-ax
is,t
he
lin
e y
�x,
and
the
lin
e
y�
�2x
�10
.Wri
te t
hre
e sy
stem
s of
equ
atio
ns
that
you
cou
ld u
se t
o fi
nd
the
thre
e ve
rtic
es o
f th
e tr
ian
gle.
(Do
not
act
ual
ly f
ind
the
vert
ices
.)
x�
0,y
�x;
x�
0,y
��
2x�
10;
y�
x,y
��
2x�
10
Rea
din
g t
he
Less
on
1.S
upp
ose
that
you
are
ask
ed t
o so
lve
the
foll
owin
g sy
stem
of
equ
atio
ns
by C
ram
er’s
Ru
le.
3x�
2y�
72x
�3y
�22
Wit
hou
t ac
tual
ly e
valu
atin
g an
y de
term
inan
ts,i
ndi
cate
wh
ich
of
the
foll
owin
g ra
tios
of
dete
rmin
ants
giv
es t
he
corr
ect
valu
e fo
r x.
B
A.
B.
C.
2.In
you
r te
xtbo
ok,t
he
stat
emen
ts o
f C
ram
er’s
Ru
le f
or t
wo
vari
able
s an
d th
ree
vari
able
ssp
ecif
y th
at t
he
dete
rmin
ant
form
ed f
rom
th
e co
effi
cien
ts o
f th
e va
riab
les
can
not
be
0.If
th
e de
term
inan
t is
zer
o,w
hat
do
you
kn
ow a
bou
t th
e sy
stem
an
d it
s so
luti
ons?
Th
e sy
stem
co
uld
be
a d
epen
den
t sy
stem
an
d h
ave
infi
nit
ely
man
yso
luti
on
s,o
r it
co
uld
be
an in
con
sist
ent
syst
em a
nd
hav
e n
o s
olu
tio
ns.
Hel
pin
g Y
ou
Rem
emb
er
3.S
ome
stud
ents
hav
e tr
oubl
e re
mem
beri
ng h
ow t
o ar
rang
e th
e de
term
inan
ts t
hat
are
used
in s
olvi
ng
a sy
stem
of
two
lin
ear
equ
atio
ns
by C
ram
er’s
Ru
le.W
hat
is
a go
od w
ay t
ore
mem
ber
this
?
Sam
ple
an
swer
:Let
Db
e th
e d
eter
min
ant
of
the
coef
ficie
nts
.Let
Dx
be t
he
dete
rmin
ant
form
ed b
y re
plac
ing
the
first
col
umn
of D
with
the
co
nst
ants
fro
m t
he
rig
ht-
han
d s
ide
of
the
syst
em,a
nd
let
Dy
be
the
det
erm
inan
tfo
rmed
by
rep
laci
ng
th
e se
con
d c
olu
mn
of
Dw
ith
th
e co
nst
ants
.Th
en
x�
and
y�
.D
y� D
Dx
� D
37
222
�
�3
2 2
�3
7
2 2
2�
3�
�
32
2
�3
3
2
2�
3�
�
72
22
�3
1 � 21 � 2
1 � 2
©G
lenc
oe/M
cGra
w-H
ill20
4G
lenc
oe A
lgeb
ra 2
Co
mm
un
icat
ion
s N
etw
ork
sT
he
diag
ram
at
the
righ
t re
pres
ents
a c
omm
un
icat
ion
s n
etw
ork
lin
kin
g fi
ve c
ompu
ter
rem
ote
stat
ion
s.T
he
arro
ws
indi
cate
th
edi
rect
ion
in
wh
ich
sig
nal
s ca
n b
e tr
ansm
itte
d an
d re
ceiv
ed b
y ea
chco
mpu
ter.
We
can
gen
erat
e a
mat
rix
to d
escr
ibe
this
net
wor
k.
A�
Mat
rix
Ais
a c
omm
unic
atio
ns n
etw
ork
for
dire
ct c
omm
unic
atio
n.S
upp
ose
you
w
ant
to s
end
a m
essa
ge f
rom
on
e co
mpu
ter
to a
not
her
usi
ng
exac
tly
one
oth
er
com
pute
r as
a r
elay
poi
nt.
It c
an b
e sh
own
th
at t
he
entr
ies
of m
atri
x A
2re
pres
ent
the
nu
mbe
r of
way
s to
sen
d a
mes
sage
fro
m o
ne
poin
t to
an
oth
er b
y go
ing
thro
ugh
a
thir
d st
atio
n.F
or e
xam
ple,
a m
essa
ge m
ay b
e se
nt
from
sta
tion
1 t
o st
atio
n 5
by
goin
g th
rou
gh s
tati
on 2
or
stat
ion
4 o
n t
he
way
.Th
eref
ore,
A2 1,
5�
2.
A2
�
For
eac
h n
etw
ork
,fin
d t
he
mat
rice
s A
and
A2 .
Th
en w
rite
th
e n
um
ber
of
way
s th
e m
essa
ges
can
be
sen
t fo
r ea
ch m
atri
x.
1.2.
3.
A:
A:
A:
9 w
ays
11 w
ays
10 w
ays
A:
A:
A:
21 w
ays
21 w
ays
14 w
ays
10
00
00
10
00
11
10
01
10
10
31
11
0
10
10
10
01
00
11
10
21
00
01
12
00
11
00
21
01
01
00
31
20
12
11
11
21
12
11
01
00
01
00
00
10
01
01
01
00
11
11
0
01
01
00
00
10
00
01
00
11
10
00
10
00
11
00
10
00
00
01
11
10
10
11
00
10
10
2
5
1 34
2 6
13
45
1
23
4
Aga
in,c
ompa
re t
he
entr
ies
of m
atri
x A
2to
th
eco
mm
un
icat
ion
s di
agra
m t
o ve
rify
th
at t
he
entr
ies
are
corr
ect.
Mat
rix
A2
repr
esen
ts u
sin
g ex
actl
y on
e re
lay.
to c
ompu
ter
j
11
11
2fr
om1
21
11
com
pute
r i1
21
11
12
04
1 1
11
12
The
ent
ry in
pos
itio
n a i
jre
pres
ents
the
num
ber
of w
ays
to s
end
a m
essa
ge f
rom
com
pute
r i
to c
ompu
ter
jdi
rect
ly.C
ompa
re t
heen
trie
s of
mat
rix
Ato
th
e di
agra
m t
o ve
rify
th
e en
trie
s.F
orex
ampl
e,th
ere
is o
ne
way
to
sen
d a
mes
sage
fro
m c
ompu
ter
3to
com
pute
r 4,
so A
3,4
�1.
A c
ompu
ter
can
not
sen
d a
mes
sage
to i
tsel
f,so
A1,
1�
0.
to c
ompu
ter
j
01
01
0fr
om0
00
11
com
pute
r i1
00
10
11
10
1 0
10
10
12
34
5
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
Answers (Lesson 4-7)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Iden
tity
an
d In
vers
e M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill20
5G
lenc
oe A
lgeb
ra 2
Lesson 4-7
Iden
tity
an
d In
vers
e M
atri
ces
Th
e id
enti
ty m
atri
x fo
r m
atri
x m
ult
ipli
cati
on i
s a
squ
are
mat
rix
wit
h 1
s fo
r ev
ery
elem
ent
of t
he
mai
n d
iago
nal
an
d ze
ros
else
wh
ere.
Iden
tity
Mat
rix
If A
is a
n n
�n
mat
rix a
nd I
is t
he id
entit
y m
atrix
,fo
r M
ult
iplic
atio
nth
en A
�I
�A
and
I�
A�
A.
If a
n n
�n
mat
rix
Ah
as a
n i
nve
rse
A�
1 ,th
en A
�A
�1
�A
�1
�A
�I.
Det
erm
ine
wh
eth
er X
�
and
Y�
ar
e in
vers
em
atri
ces.
Fin
d X
�Y
.
X�
Y�
�
�
or
Fin
d Y
�X
.
Y�
X�
�
�
or
Sin
ce X
�Y
�Y
�X
�I,
Xan
d Y
are
inve
rse
mat
rice
s.
Det
erm
ine
wh
eth
er e
ach
pai
r of
mat
rice
s ar
e in
vers
es.
1.an
d 2.
and
3.an
d
yes
yes
no
4.an
d 5.
and
6.an
d
yes
no
yes
7.an
d 8.
and
9.an
d
no
yes
no
10.
and
11.
and
12.
and
no
yes
yes
�5
3
7�
44
3 7
5
5�
2 �
177
7
2 1
75
3
2 �
4�
33
2 4
�6
�7 2�
��3 2�
1
�2
37
24
�3
4
2�
�5 2� 5
8 4
6
��1 5�
�� 11 0�
� 13 0�
� 11 0�
42
6�
2
�2
1
�1 21 �
��5 2�
5
2 1
14
12
38
4�
1 5
3�
411
3�
88
11
34
2
3 �
1�
22
3 5
�1
2
�1
��5 2�
�3 2� 3
2 5
4
4�
5 �
34
45
34
10
01
21
�20
12 �
12
�35
�35
�20
�21
7
4 1
06
3
�2
�5
�7 2�
10
01
21
�20
�14
�14
3
0 �
30�
20 �
21
3
�2
�5
�7 2�
74
10
6
3
�2
�
5�7 2�
74
10
6Ex
ampl
eEx
ampl
e
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill20
6G
lenc
oe A
lgeb
ra 2
Fin
d In
vers
e M
atri
ces
Inve
rse
of
a 2
�2
Mat
rix
The
inve
rse
of a
mat
rix A
�
is
A�
1�
, w
here
ad
�bc
�0.
If a
d�
bc�
0,th
e m
atri
x do
es n
ot h
ave
an i
nve
rse.
Fin
d t
he
inve
rse
of N
�.
Fir
st f
ind
the
valu
e of
th
e de
term
inan
t.
�7
�4
�3
Sin
ce t
he
dete
rmin
ant
does
not
equ
al 0
,N�
1ex
ists
.
N�
1�
��
Ch
eck
:
NN
�1
�
� �
�
N�
1 N�
�
�
�
Fin
d t
he
inve
rse
of e
ach
mat
rix,
if i
t ex
ists
.
1.2.
3.
no
inve
rse
exis
ts
4.5.
6.
��1 2�
no
inve
rse
exis
ts
4�
2�
10�
81 � 12
8
�5
�10
6
8
2 1
04
36
48
6
5 1
08
30
10
20
40
1� 10
001
�1
01
40
�10
�
2030
1
1 0
12
412
84
10
01
�7 3�
� �4 3�
�2 3��
�2 3�
��1 34 �
� �1 34 �
��4 3�
� �7 3�
72
21
�1 3�
��2 3�
��2 3�
�7 3�
10
01
�7 3��
�4 3��
�1 34 ��
�1 34 �
�2 3��
�2 3��
�4 3��
�7 3�
�1 3�
��2 3�
��2 3�
�7 3�
72
21
�1 3�
��2 3�
��2 3�
�7 3�
1
�2
�2
71 � 3
d
�b
�c
a1
� ad�
bc
72
21
72
21
d
�b
�c
a1
� ad�
bc
ab
c
d
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Iden
tity
an
d In
vers
e M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A21 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-7)
Skil
ls P
ract
ice
Iden
tity
an
d In
vers
e M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill20
7G
lenc
oe A
lgeb
ra 2
Lesson 4-7
Det
erm
ine
wh
eth
er e
ach
pai
r of
mat
rice
s ar
e in
vers
es.
1.X
�,Y
�ye
s2.
P�
,Q�
yes
3.M
�,N
�n
o4.
A�
,B�
yes
5.V
�,W
�ye
s6.
X�
,Y�
yes
7.G
�,H
�ye
s8.
D�
,E�
no
Fin
d t
he
inve
rse
of e
ach
mat
rix,
if i
t ex
ists
.
9.�
10.
�
11.
no
inve
rse
exis
ts12
.
13.
14.
no
inve
rse
exis
ts
15.
16.
�
17.
�18
.n
o in
vers
e ex
ists
19.
�20
.2
00
21 � 4
20
02
�
8�
8�
1010
1
� 160
10
8 1
0�
8
10
8
54
07
70
1 � 49
0�
7 �
70
2
�5
�1
�4
1 � 13�
45
1
2�
11
�1
�1
1 � 2�
1�
1
1�
1
3
6 �
1�
2
31
�3
11 � 6
1�
1 3
3
0
4�
6�
21 � 24
�2
�4
6
09
3 6
2
2
�1
�3
11
1 3
2
0�
2�
40
1 � 80
2 4
0
�0.
125
�0.
125
�0.
125
�0.
125
�4
�4
�4
4
� 12 1�
� 13 1�
�� 11 1�
� 14 1�
4�
3 1
2
��1 3�
�2 3�
�1 6�
�1 6�
�1
4
12
0�
�1 7�
�1 7�0
0
7 �
70
2�
5 1
�2
�2
5 �
12
�1
0
0�
3�
10
0
3
�1
3
1�
22
3 1
1
10
�1
11
0 1
1
©G
lenc
oe/M
cGra
w-H
ill20
8G
lenc
oe A
lgeb
ra 2
Det
erm
ine
wh
eth
er e
ach
pai
r of
mat
rice
s ar
e in
vers
es.
1.M
�,N
�n
o2.
X�
,Y�
yes
3.A
�,B
�ye
s4.
P�
,Q�
yes
Det
erm
ine
wh
eth
er e
ach
sta
tem
ent
is t
rue
or f
als
e.
5.A
ll s
quar
e m
atri
ces
hav
e m
ult
ipli
cati
ve i
nve
rses
.fa
lse
6.A
ll s
quar
e m
atri
ces
hav
e m
ult
ipli
cati
ve i
den
titi
es.
tru
e
Fin
d t
he
inve
rse
of e
ach
mat
rix,
if i
t ex
ists
.
7.8.
9.�
10.
11.
12.
no
inve
rse
exis
ts
GEO
MET
RYF
or E
xerc
ises
13–
16,u
se t
he
figu
re a
t th
e ri
ght.
13.W
rite
th
e ve
rtex
mat
rix
Afo
r th
e re
ctan
gle.
14.U
se m
atri
x m
ult
ipli
cati
on t
o fi
nd
BA
if B
�.
15.G
raph
th
e ve
rtic
es o
f th
e tr
ansf
orm
ed t
rian
gle
on t
he
prev
iou
s gr
aph
.D
escr
ibe
the
tran
sfor
mat
ion
.d
ilati
on
by
a sc
ale
fact
or
of
1.5
16.M
ake
a co
nje
ctu
re a
bou
t w
hat
tra
nsf
orm
atio
n B
�1
desc
ribe
s on
a c
oord
inat
e pl
ane.
dila
tio
n b
y a
scal
e fa
cto
r o
f
17.C
OD
ESU
se t
he
alph
abet
tab
le b
elow
an
d th
e in
vers
e of
cod
ing
mat
rix
C�
to
deco
de t
his
mes
sage
:19
|14|11
|13|11
|22|55
|65|57
|60|2
|1|52
|47|33
|51|56
|55.
CH
EC
K_Y
OU
R_A
NS
WE
RS
CO
DE
A1
B2
C3
D4
E5
F6
G7
H8
I9
J10
K11
L12
M13
N14
O15
P16
Q17
R18
S19
T20
U21
V22
W23
X24
Y25
Z26
–0
12
21
2 � 3
1.5
67.
53
3
61.
5�
1.5
1.5
0
01.
5
14
52
24
1�
1
( 6, 6
)
( 7.5
, 1.5
)
( 3, –
1.5)
( 1.5
, 3)
x
y
O
( 2, –
1)
( 1, 2
)
( 4, 4
)
( 5, 1
)
46
69
1
5�
32
1 � 172
�5
31
3�
51
21 � 11
2
5 �
13
�7
�3
�4
�1
1 � 5�
13
4
�7
5
0�
32
1 � 102
0 3
5�
3�
5
44
1 � 8
45
�4
�3
� 13 4��1 7�
�1 7�
�3 7�
6
�2
�2
3
�1 5��
� 11 0�
�2 5�� 13 0�
3
1 �
42
32
53
�3
2
5�
3�
21
3
�2
21
32Pra
ctic
e (
Ave
rag
e)
Iden
tity
an
d In
vers
e M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
Answers (Lesson 4-7)
Readin
g t
o L
earn
Math
em
ati
csId
enti
ty a
nd
Inve
rse
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill20
9G
lenc
oe A
lgeb
ra 2
Lesson 4-7
Pre-
Act
ivit
yH
ow a
re i
nve
rse
mat
rice
s u
sed
in
cry
pto
grap
hy?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-7
at
the
top
of p
age
195
in y
our
text
book
.
Ref
er t
o th
e co
de t
able
giv
en i
n t
he
intr
odu
ctio
n t
o th
is l
esso
n.S
upp
ose
that
you
rec
eive
a m
essa
ge c
oded
by
this
sys
tem
as
foll
ows:
1612
51
195
25
1325
618
95
144.
Dec
ode
the
mes
sage
.P
leas
e b
e m
y fr
ien
d.
Rea
din
g t
he
Less
on
1.In
dica
te w
het
her
eac
h o
f th
e fo
llow
ing
stat
emen
ts i
s tr
ue
or f
alse
.
a.E
very
ele
men
t of
an
ide
nti
ty m
atri
x is
1.
fals
e
b.
Th
ere
is a
3 �
2 id
enti
ty m
atri
x.fa
lse
c.T
wo
mat
rice
s ar
e in
vers
es o
f ea
ch o
ther
if
thei
r pr
odu
ct i
s th
e id
enti
ty m
atri
x.tr
ue
d.
If M
is a
mat
rix,
M�
1re
pres
ents
th
e re
cipr
ocal
of
M.
fals
e
e.N
o 3
�2
mat
rix
has
an
in
vers
e.tr
ue
f.E
very
squ
are
mat
rix
has
an
in
vers
e.fa
lse
g.If
th
e tw
o co
lum
ns
of a
2 �
2 m
atri
x ar
e id
enti
cal,
the
mat
rix
does
not
hav
e an
inve
rse.
tru
e
2.E
xpla
in h
ow t
o fi
nd
the
inve
rse
of a
2 �
2 m
atri
x.D
o n
ot u
se a
ny
spec
ial
mat
hem
atic
alsy
mbo
ls i
n y
our
expl
anat
ion
.
Sam
ple
an
swer
:F
irst
fin
d t
he
det
erm
inan
t o
f th
e m
atri
x.If
it is
zer
o,t
hen
the
mat
rix
has
no
inve
rse.
If t
he
det
erm
inan
t is
no
t ze
ro,f
orm
a n
ewm
atri
x as
fo
llow
s.In
terc
han
ge
the
top
left
an
d b
ott
om
rig
ht
elem
ents
.C
han
ge
the
sig
ns
but
no
t th
e p
osi
tio
ns
of
the
oth
er t
wo
ele
men
ts.
Mu
ltip
ly t
he
resu
ltin
g m
atri
x by
th
e re
cip
roca
l of
the
det
erm
inan
t o
f th
eo
rig
inal
mat
rix.
Th
e re
sult
ing
mat
rix
is t
he
inve
rse
of
the
ori
gin
al m
atri
x.
Hel
pin
g Y
ou
Rem
emb
er
3.O
ne
way
to
rem
embe
r so
met
hin
g is
to
expl
ain
it
to a
not
her
per
son
.Su
ppos
e th
at y
ou a
rest
udy
ing
wit
h a
cla
ssm
ate
wh
o is
hav
ing
trou
ble
rem
embe
rin
g h
ow t
o fi
nd
the
inve
rse
ofa
2 �
2 m
atri
x.H
e re
mem
bers
how
to
mov
e el
emen
ts a
nd
chan
ge s
ign
s in
th
e m
atri
x,bu
t th
inks
th
at h
e sh
ould
mu
ltip
ly b
y th
e de
term
inan
t of
th
e or
igin
al m
atri
x.H
ow c
anyo
u h
elp
him
rem
embe
r th
at h
e m
ust
mu
ltip
ly b
y th
e re
cipr
ocal
of t
his
det
erm
inan
t?
Sam
ple
an
swer
:If
th
e d
eter
min
ant
of
the
mat
rix
is 0
,its
rec
ipro
cal i
su
nd
efin
ed.T
his
ag
rees
wit
h t
he
fact
th
at if
th
e d
eter
min
ant
of
a m
atri
x is
0,th
e m
atri
x d
oes
no
t h
ave
an in
vers
e.
©G
lenc
oe/M
cGra
w-H
ill21
0G
lenc
oe A
lgeb
ra 2
Per
mu
tati
on
Mat
rice
sA
per
mu
tati
on m
atri
x is
a s
quar
e m
atri
x in
wh
ich
ea
ch r
ow a
nd
each
col
um
n h
as o
ne
entr
y th
at i
s 1.
All
th
e ot
her
en
trie
s ar
e 0.
Fin
d th
e in
vers
e of
ape
rmu
tati
on m
atri
x in
terc
han
gin
g th
e ro
ws
and
colu
mn
s.F
or e
xam
ple,
row
1 i
s in
terc
han
ged
wit
hco
lum
n 1
,row
2 i
s in
terc
han
ged
wit
h c
olu
mn
2.
Pis
a 4
�4
perm
uta
tion
mat
rix.
P�
1is
th
e in
vers
e of
P.
Sol
ve e
ach
pro
ble
m.
1.T
her
e is
just
on
e 2
�2
perm
uta
tion
2.
Fin
d th
e in
vers
e of
th
e m
atri
x yo
u w
rote
m
atri
x th
at i
s n
ot a
lso
an i
den
tity
in
Exe
rcis
e 1.
Wh
at d
o yo
u n
otic
e?m
atri
x.W
rite
th
is m
atri
x.
Th
e tw
o m
atri
ces
are
the
sam
e.
3.S
how
th
at t
he
two
mat
rice
s in
Exe
rcis
es 1
an
d 2
are
inve
rses
.
�
4.W
rite
th
e in
vers
e of
th
is m
atri
x.
B�
B�
1�
5.U
se B
�1
from
pro
blem
4.V
erif
y th
at B
and
B�
1ar
e in
vers
es.
�
6.P
erm
uta
tion
mat
rice
s ca
n b
e u
sed
to w
rite
an
d de
ciph
er c
odes
.To
see
how
th
is i
s do
ne,
use
th
e m
essa
ge m
atri
x M
and
mat
rix
Bfr
om p
robl
em 4
.Fin
d m
atri
x C
so t
hat
Ceq
ual
s th
e pr
odu
ct M
B.U
se t
he
rule
s be
low
.
0 tim
es a
lette
r �
0
1 tim
es a
lette
r �
the
sam
e le
tter
0 pl
us a
lette
r �
the
sam
e le
tter
7.N
ow f
ind
the
prod
uct
CB
�1 .
Wh
at d
o yo
u n
otic
e?
�
�
Mu
ltip
lyin
g M
by B
enco
des
th
e m
essa
ge.
To d
ecip
her
,mu
ltip
ly b
y B
�1 .
SH
E
SA
W
HI
M
01
00
01
10
0
HE
S
AW
S
I
MH
01
00
01
10
0
0 �
0 �
0 �
0 �
1 �
10
�1
�0
�0
�1
�0
0 �
0 �
0 �
1 �
1 �
01
�0
�0
�0
�0
�1
1 �
1 �
0 �
0 �
0 �
01
�0
�0
�1
�0
�0
0 �
0 �
1 �
0 �
0 �
10
�1
�1
�0
�0
�0
0 �
0 �
1 �
1 �
0 �
0
01
00
01
10
0
00
11
00
01
0
10
01
0 �
1 �
1 �
10
�1
�1
�0
1 �
0 �
0 �
11
�1
�0
�0
01
10
01
10
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
P�
P�
1�
00
01
01
00
10
00
00
10
00
10
01
00
00
01
10
00
M�
C�
HE
S
AW
S
I
MH
SH
E
SA
W
HI
M
© Glencoe/McGraw-Hill A23 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-8)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Usi
ng
Mat
rice
s to
So
lve
Sys
tem
s o
f E
qu
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-8
4-8
©G
lenc
oe/M
cGra
w-H
ill21
1G
lenc
oe A
lgeb
ra 2
Lesson 4-8
Wri
te M
atri
x Eq
uat
ion
sA
mat
rix
equ
atio
nfo
r a
syst
em o
f eq
uat
ion
s co
nsi
sts
of t
he
prod
uct
of
the
coef
fici
ent
and
vari
able
mat
rice
s on
th
e le
ft a
nd
the
con
stan
t m
atri
x on
th
eri
ght
of t
he
equ
als
sign
.
Wri
te a
mat
rix
equ
atio
n f
or e
ach
sys
tem
of
equ
atio
ns.
a.3x
�7y
�12
x�
5y�
�8
Det
erm
ine
the
coef
fici
ent,
vari
able
,an
dco
nst
ant
mat
rice
s.
��
12
�8
x
y
3�
7 1
5
b.
2x�
y�
3z�
�7
x�
3y�
4z�
157x
�2y
�z
��
28
� �
�
7
15
�28
x
y
z
2�
13
13
�4
72
1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Wri
te a
mat
rix
equ
atio
n f
or e
ach
sys
tem
of
equ
atio
ns.
1.2x
�y
�8
2.4x
�3y
�18
3.7x
�2y
�15
5x�
3y�
�12
x�
2y�
123x
�y
��
10
� �
�
�
� �
4.4x
�6y
�20
5.5x
�2y
�18
6.3x
�y
�24
3x�
y�
8�0
x�
�4y
�25
3y�
80 �
2x
� �
�
�
� �
7.2x
�y
�7z
�12
8.5x
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y�
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18x
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12
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9.4x
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100
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y�
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2x�
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72
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11.2
x�
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812
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45 �
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3y�
z�
603x
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120
� �
�
�
45
60
1
20
x
y
z
3�
21
23
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1�
42
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8
40
89
x
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23
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1�
25
35
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27
48
72
x
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37
21
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23
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0
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8
x
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35
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2
32
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18
12
x
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z
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17
13
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24
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12
15
25
x
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21
75
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31
2�
6
24
80
x
y
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12
31
82
5x
y
5
21
42
0�
8x
y
4
�6
31
15
�
10
x
y
7�
23
11
81
2x
y
4
�3
12
8
�12
x
y
2
15
�3
©G
lenc
oe/M
cGra
w-H
ill21
2G
lenc
oe A
lgeb
ra 2
Solv
e Sy
stem
s o
f Eq
uat
ion
sU
se i
nve
rse
mat
rice
s to
sol
ve s
yste
ms
of e
quat
ion
sw
ritt
en a
s m
atri
x eq
uat
ion
s.
So
lvin
g M
atri
x E
qu
atio
ns
If A
X�
B,
then
X�
A�
1 B,
whe
re A
is t
he c
oeffi
cien
t m
atrix
, X
is t
he v
aria
ble
mat
rix,
and
Bis
the
con
stan
t m
atrix
.
Sol
ve
�.
In t
he
mat
rix
equ
atio
n A
�,X
�,a
nd
B�
.
Ste
p 1
Fin
d th
e in
vers
e of
th
e co
effi
cien
t m
atri
x.
A�
1�
or
.
Ste
p 2
Mu
ltip
ly e
ach
sid
e of
th
e m
atri
x eq
uat
ion
by
the
inve
rse
mat
rix.
� �
��
Mul
tiply
eac
h si
de b
y A
�1 .
� �
Mul
tiply
mat
rices
.
�S
impl
ify.
Th
e so
luti
on i
s (2
,�2)
.
Sol
ve e
ach
mat
rix
equ
atio
n o
r sy
stem
of
equ
atio
ns
by
usi
ng
inve
rse
mat
rice
s.
1.�
�2.
� �
3.�
�
(5,�
3)n
o s
olu
tio
n(1
3,�
18)
4.�
�5.
� �
6.�
�
��,�
�(5
7,�
31)
��,
�7.
4x�
2y�
228.
2x�
y�
29.
3x�
4y�
126x
�4y
��
2x
�2y
�46
5x�
8y�
�8
(3,�
5)(1
0,18
)(3
2,�
21)
10.x
�3y
��
511
.5x
�4y
�5
12.3
x�
2y�
52x
�7y
�8
9x�
8y�
0x
�4y
�20
(�59
,18)
�,
�(�
2,�
5.5)
45 � 7610 � 19
15 � 79 � 7
3 � 21 � 4
3
�6
x
y
12
3�
1�
15
6
x
y
36
59
4
�8
x
y
2�
3 2
5
3
�7
x
y
32
54
16
12
x
y
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�8
6
12
�2
18
x
y
2
4 3
�1
2
�2
x
y
16
�
16
1 � 8x
y
1
0 0
1
6
4
4
�2
�6
51 � 8
x
y
52
64
4
�2
�6
51 � 8
4
�2
�6
51 � 8
4
�2
�6
51
� 20 �
12
6
4
x
y
52
64
6
4
x
y
52
64
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Usi
ng
Mat
rice
s to
So
lve
Sys
tem
s o
f E
qu
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-8
4-8
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A24 Glencoe Algebra 2
Answers (Lesson 4-8)
Skil
ls P
ract
ice
Usi
ng
Mat
rice
s to
So
lve
Sys
tem
s o
f E
qu
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-8
4-8
©G
lenc
oe/M
cGra
w-H
ill21
3G
lenc
oe A
lgeb
ra 2
Lesson 4-8
Wri
te a
mat
rix
equ
atio
n f
or e
ach
sys
tem
of
equ
atio
ns.
1.x
�y
�5
2.3a
�8b
�16
2x�
y�
14a
�3b
�3
� �
�
�
3.m
�3n
��
34.
2c�
3d�
64m
�3n
�6
3c�
4d�
7
� �
�
�
5.r
�s
�1
6.x
�y
�5
2r�
3s�
123x
�2y
�10
� �
�
�
7.6x
�y
�2z
��
48.
a�
b�
c�
5�
3x�
2y�
z�
103a
�2b
�c
�0
x�
y�
z�
32a
�3b
�8
� �
�
�
Sol
ve e
ach
mat
rix
equ
atio
n o
r sy
stem
of
equ
atio
ns
by
usi
ng
inve
rse
mat
rice
s.
9.�
�(2
,�3)
10.
��
(3,�
2)
11.
��
(3,�
2)12
.�
�(3
,2)
13.
��
�7,�
14.
��
�1,�
15.p
�3q
�6
16.�
x�
3y�
22p
�3q
��
6(0
,�2)
�4x
�5y
�1
(1,�
1)
17.2
m�
2n�
�8
18.�
3a�
b�
�9
6m�
4n�
�18
(�1,
�3)
5a�
2b�
14(4
,3)
5 � 31
5
2m
n
56
12
�6
1 � 32
5 1
2c
d
3
12
2�
6
15
23
m
n
7�
3 5
4�
1
7a
b
5
8 3
1
6
�3
x
y
43
13
�7
�1
w
z
13
43
5
0
8
a
b
c
1�
11
32
�1
23
0
�4
10
3
x
y
z
6
�1
2�
32
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1
11
5
10
x
y
11
32
1
12
r
s
1�
12
3
6
7
c
d
23
3�
4�
3
6m
n
13
43
16
3
a
b
38
43
5
1
x
y
11
2�
1
©G
lenc
oe/M
cGra
w-H
ill21
4G
lenc
oe A
lgeb
ra 2
Wri
te a
mat
rix
equ
atio
n f
or e
ach
sys
tem
of
equ
atio
ns.
1.�
3x�
2y�
92.
6x�
2y�
�2
5x�
3y�
�13
3x�
3y�
10
� �
�
�
3.2a
�b
�0
4.r
�5s
�10
3a�
2b�
�2
2r�
3s�
7
� �
�
�
5.3x
�2y
�5z
�3
6.2m
�n
�3p
��
5x
�y
�4z
�2
5m�
2n�
2p�
8�
2x�
2y�
7z�
�5
3m�
3n�
5p�
17
� �
�
�
Sol
ve e
ach
mat
rix
equ
atio
n o
r sy
stem
of
equ
atio
ns
by
usi
ng
inve
rse
mat
rice
s.
7.�
�(2
,�4)
8.�
�(5
,1)
9.�
�(3
,�5)
10.
��
(1,7
)
11.
��
��4,
�12
.�
���
,�
13.2
x�
3y�
514
.8d
�9f
�13
3x�
2y�
1(1
,1)
�6d
�5f
��
45(5
,�3)
15.5
m�
9n�
1916
.�4j
�9k
��
82m
�n
��
20(�
7,6)
6j�
12k
��
5�
,��
17.A
IRLI
NE
TIC
KET
SL
ast
Mon
day
at 7
:30
A.M
.,an
air
lin
e fl
ew 8
9 pa
ssen
gers
on
aco
mm
ute
r fl
igh
t fr
om B
osto
n t
o N
ew Y
ork.
Som
e of
th
e pa
ssen
gers
pai
d $1
20 f
or t
hei
rti
cket
s an
d th
e re
st p
aid
$230
for
th
eir
tick
ets.
Th
e to
tal
cost
of
all
of t
he
tick
ets
was
$14,
200.
How
man
y pa
ssen
gers
bou
ght
$120
tic
kets
? H
ow m
any
bou
ght
$230
tic
kets
?57
;32
18.N
UTR
ITIO
NA
sin
gle
dose
of
a di
etar
y su
pple
men
t co
nta
ins
0.2
gram
of
calc
ium
an
d 0.
2 gr
am o
f vi
tam
in C
.A s
ingl
e do
se o
f a
seco
nd
diet
ary
supp
lem
ent
con
tain
s 0.
1 gr
am
of c
alci
um
an
d 0.
4 gr
am o
f vi
tam
in C
.If
a pe
rson
wan
ts t
o ta
ke 0
.6 g
ram
of
calc
ium
an
d1.
2 gr
ams
of v
itam
in C
,how
man
y do
ses
of e
ach
su
pple
men
t sh
ould
sh
e ta
ke?
2 d
ose
s o
f th
e fi
rst
sup
ple
men
t an
d 2
do
ses
of
the
seco
nd
su
pp
lem
ent
2 � 31 � 2
1 � 31 � 4
�1
�1
y
z
8
3 1
26
1 � 2
17
�26
r
s
�
42
7
4
16
34
c
d
�5
3
64
12
�
11
a
b
�1
�3
3
4
�7
10
x
y
�2
3
15
0
�2
g
h
21
32
�5
8
17
m
n
p
21
�3
52
�2
3�
35
3
2
�5
x
y
z
3
�2
5
11
�4
�2
27
10
7
r
s
15
2�
3
0�
2a
b
2
13
2
�2
10
x
y
6�
23
3
9�
13
x
y
�3
2
5�
3Pra
ctic
e (
Ave
rag
e)
Usi
ng
Mat
rice
s to
So
lve
Sys
tem
s o
f E
qu
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-8
4-8
© Glencoe/McGraw-Hill A25 Glencoe Algebra 2
An
swer
s
Answers (Lesson 4-8)
Readin
g t
o L
earn
Math
em
ati
csU
sin
g M
atri
ces
to S
olv
e S
yste
ms
of
Eq
uat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-8
4-8
©G
lenc
oe/M
cGra
w-H
ill21
5G
lenc
oe A
lgeb
ra 2
Lesson 4-8
Pre-
Act
ivit
yH
ow a
re i
nve
rse
mat
rice
s u
sed
in
pop
ula
tion
eco
logy
?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-8
at
the
top
of p
age
202
in y
our
text
book
.
Wri
te a
2 �
2 m
atri
x th
at s
um
mar
izes
th
e in
form
atio
n g
iven
in
th
ein
trod
uct
ion
abo
ut
the
food
an
d te
rrit
ory
requ
irem
ents
for
th
e tw
o sp
ecie
s.
Rea
din
g t
he
Less
on
1.a.
Wri
te a
mat
rix
equ
atio
n f
or t
he
foll
owin
g sy
stem
of
equ
atio
ns.
3x�
5y�
10�
�
2x�
4y�
�7
b.
Exp
lain
how
to
use
th
e m
atri
x eq
uat
ion
you
wro
te a
bove
to
solv
e th
e sy
stem
.Use
as
few
mat
hem
atic
al s
ymbo
ls i
n y
our
expl
anat
ion
as
you
can
.Do
not
act
ual
ly s
olve
th
esy
stem
.
Sam
ple
an
swer
:F
ind
th
e in
vers
e o
f th
e 2
�2
mat
rix
of
coef
fici
ents
.M
ult
iply
th
is in
vers
e by
th
e 2
�1
mat
rix
of
con
stan
ts,w
ith
th
e 2
�2
mat
rix
on
th
e le
ft.T
he
pro
du
ct w
ill b
e a
2 �
1 m
atri
x.T
he
nu
mb
er in
the
firs
t ro
w w
ill b
e th
e va
lue
of
x,an
d t
he
nu
mb
er in
th
e se
con
d r
ow
will
be
the
valu
e o
f y.
2.W
rite
a s
yste
m o
f eq
uat
ion
s th
at c
orre
spon
ds t
o th
e fo
llow
ing
mat
rix
equ
atio
n.
� �
3x�
2y�
4z�
�2
2x�
y�
65y
�6z
��
4
Hel
pin
g Y
ou
Rem
emb
er
3.S
ome
stu
den
ts h
ave
trou
ble
rem
embe
rin
g h
ow t
o se
t u
p a
mat
rix
equ
atio
n t
o so
lve
asy
stem
of
lin
ear
equ
atio
ns.
Wh
at i
s an
eas
y w
ay t
o re
mem
ber
the
orde
r in
wh
ich
to
wri
teth
e th
ree
mat
rice
s th
at m
ake
up
the
equ
atio
n?
Sam
ple
an
swer
:Ju
st r
emem
ber
“C
VC
”fo
r “c
oef
fici
ents
,var
iab
les,
con
stan
ts.”
Th
e va
riab
le m
atri
x is
on
th
e le
ft s
ide
of
the
equ
als
sig
n,j
ust
as t
he
vari
able
s ar
e in
th
e sy
stem
of
linea
r eq
uat
ion
s.T
he
con
stan
tm
atri
x is
on
th
e ri
gh
t si
de
of
the
equ
als
sig
n,j
ust
as
the
con
stan
ts a
re in
the
syst
em o
f lin
ear
equ
atio
ns.
�2
6
�4
x
y
z
32
�4
2�
10
05
6
10
�7
x
y
35
2�
4
140
500
120
400
©G
lenc
oe/M
cGra
w-H
ill21
6G
lenc
oe A
lgeb
ra 2
Pro
per
ties
of
Mat
rice
sC
ompu
tin
g w
ith
mat
rice
s is
dif
fere
nt
from
com
puti
ng
wit
h r
eal
nu
mbe
rs.S
tate
d be
low
are
som
e pr
oper
ties
of
the
real
nu
mbe
r sy
stem
.Are
th
ese
also
tru
e fo
r m
atri
ces?
In
th
e pr
oble
ms
on t
his
pag
e,yo
u w
ill
inve
stig
ate
this
qu
esti
on.
For
all
rea
l n
um
bers
aan
d b,
ab�
0 if
an
d on
ly i
f a
�0
or b
�0.
Mu
ltip
lica
tion
is
com
mu
tati
ve.F
or a
ll r
eal
nu
mbe
rs a
and
b,ab
�ba
.
Mu
ltip
lica
tion
is
asso
ciat
ive.
For
all
rea
l n
um
bers
a,b
,an
d c,
a(bc
) �
(ab)
c.
Use
th
e m
atri
ces
A,B
,an
d C
for
the
pro
ble
ms.
Wri
te w
het
her
eac
h
stat
emen
t is
tru
e.A
ssu
me
that
a 2
-by-
2 m
atri
x is
th
e 0
mat
rix
if a
nd
on
ly i
f al
l of
its
ele
men
ts a
re z
ero.
A�
B
�
C�
1.A
B�
0n
o2.
AC
�0
no
3.B
C�
0ye
s
AB
�
AB
�
AB
�
4.A
B�
BA
no
5.A
C�
CA
no
6.B
C�
CB
no
BA
�
CA
�
AB
�
So
,AB
� B
A.
So
,AC
� C
A.
So
,BC
� C
B.
7.A
(BC
) �
(AB
)Cye
s8.
B(C
A)
�(B
C)A
yes
9.B
(AC
) �
(BA
)Cye
sB
oth
pro
du
cts
equ
alB
oth
pro
du
cts
equ
alB
oth
pro
du
cts
equ
al
..
.
10.W
rite
a s
tate
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© Glencoe/McGraw-Hill A26 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
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Chapter 4 Assessment Answer Key Form 1 Form 2APage 217 Page 218 Page 219
(continued on the next page)
© Glencoe/McGraw-Hill A27 Glencoe Algebra 2
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: 0
B
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A
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C
0
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C
B
C
C
A
B
D
Chapter 4 Assessment Answer Key Form 2A (continued) Form 2BPage 220 Page 221 Page 222
An
swer
s
© Glencoe/McGraw-Hill A28 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
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(3.5, �2)
50 units2
174
29
R�(4, �2), S�(0, �2),T�(�6, 1), U�(�2, 1)
A�(�1, �5),B�(5, �7), C�(2, 2)
[18 �12 22 48]
impossible
(3, �4)
Chapter 4 Assessment Answer Key Form 2CPage 223 Page 224
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© Glencoe/McGraw-Hill A29 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
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B: 2ac
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impossible
Chapter 4 Assessment Answer Key Form 2DPage 225 Page 226
An
swer
s
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© Glencoe/McGraw-Hill A30 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
no solution
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(�2, 5)
Chapter 4 Assessment Answer Key Form 3Page 227 Page 228
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Chapter 4 Assessment Answer KeyPage 229, Open-Ended Assessment
Scoring Rubric
© Glencoe/McGraw-Hill A31 Glencoe Algebra 2
Score General Description Specific Criteria
• Shows thorough understanding of the concepts of adding,subtracting, and multiplying matrices; multiplying by ascalar; transforming geometric figures using matrices;evaluating determinants; solving systems of equationsusing Cramer’s Rule; identifying and finding inversematrices; and using inverse matrices to solve matrixequations.
• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Goes beyond requirements of some or all problems.
• Shows an understanding of the concepts of adding,subtracting, and multiplying matrices; multiplying by a scalar;transforming geometric figures using matrices; evaluatingdeterminants; solving systems of equations using Cramer’sRule; identifying and finding inverse matrices; and usinginverse matrices to solve matrix equations.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.
• Shows an understanding of most of the concepts ofadding, subtracting, and multiplying matrices; multiplyingby a scalar; transforming geometric figures using matrices;evaluating determinants; solving systems of equationsusing Cramer’s Rule; identifying and finding inversematrices; and using inverse matrices to solve matrixequations.
• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies the requirements of most of the problems.
• Final computation is correct.• No written explanations or work is shown to substantiate
the final computation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of most of the concepts ofadding, subtracting, and multiplying matrices; multiplyingby a scalar; transforming geometric figures using matrices;evaluating determinants; solving systems of equationsusing Cramer’s Rule; identifying and finding inversematrices; and using inverse matrices to solve matrixequations.
• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Does not satisfy requirements of problems.• No answer may be given.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
An
swer
s
© Glencoe/McGraw-Hill A32 Glencoe Algebra 2
Chapter 4 Assessment Answer Key Page 229, Open-Ended Assessment
Sample Answers
1. Student responses should include:1a. A and B must have the same
dimensions, so m � j and n � k.1b. Inner dimensions must be equal. Thus,
n � j for AB and k � m for BA.1c. The determinant of A exists only if A is
a square matrix, so m � n.1d. Matrix size for scalar multiplication is
irrelevant, so there are no restrictionson j and k.
1e. Only square matrices (potentially) haveinverses, so m � n if A has an inverse.
2a. Student matrices must be of the form
V � � �, where P1(x1, y1),
P2(x2, y2), and P3(x3, y3) are the points,no two in the same quadrant, which thestudent has chosen. All students shouldstate that V is a 2 � 3 matrix.
2b. Student matrices must be of the formT � V, which represents a translation ofthe original triangle 3 units right and 4units down.
T � V � � �2c. Student matrices must be of the form
3V, which represents a dilation of theoriginal triangle which triples thelength of each side.
3V � � �2d. Student matrices must be of the form
RV, which represents a reflection of theoriginal triangle over the y-axis.
RV � � �
2e. Student matrices must be of the formCV, which represents a 90�counterclockwise rotation of the originaltriangle about the origin.
CV � � �2f. Students should indicate that I is the
identity matrix, so that IV � V. Thismeans that the image and the preimageof the triangle are exactly the same inall respects.
3a. Sample answer: Expansion using thebottom (3rd) row would be easiest sincethe elements are 1, 0 and �1.
3b. Student responses should indicate thereason for choosing one method over theother.
3c. Students should correctly apply thechosen method to arrive at the solutiona � �5.
4a. 3c � 2v � 85 2c � v � 50(Variables may vary); c: the cost of oneCD, v: the cost of one video
4b. Students should correctly applyCramer’s Rule to determine that c � 15and v � 20, meaning that the cost of oneCD is $15 and the cost of one video is$20.
4c. � � � � � � � �; Students should
demonstrate the correct method ofsolving the equation using inversematrices, and again conclude that c � 15and v � 20.
4d. Student responses should indicate thereason for choosing one method over theother.
8550
cv
3 22 1
�y1 �y2 �y3x1 x2 x3
�x1 �x2 �x3y1 y2 y3
3x1 3x2 3x33y1 3y2 3y3
x1 � 3 x2 � 3 x3 � 3y1 � 4 y2 � 4 y3 � 4
x1 x2 x3y1 y2 y3
In addition to the scoring rubric found on page A31, the following sample answers may be used as guidance in evaluating open-ended assessment items.
© Glencoe/McGraw-Hill A33 Glencoe Algebra 2
1. Cramer’s Rule
2. square matrix
3. transformation
4. matrix
5. reflection
6. Expansion byminors
7. minor
8. isometry
9. vertex matrix
10. dilation
11. Sample answer:An identity matrix is a square matrixin which theelements on themain diagonal areall ones and all theother elements arezeroes.
12. Sample answer:A column matrix isa matrix that hasexactly one column.
1.
2.
3.
4.
5.
Quiz (Lessons 4–3 and 4–4)
Page 231
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
Quiz (Lessons 4–7 and 4–8)
Page 232
1.
2.
3.
4.
5. (4, 5)
no inverse exists
yes
(�2, 3, 1)
(�1, 7)
24 units2
52
7
X�(3, 2), Y�(2, �3),Z�(�5, 4)
A(�2, �4), D�(1, 6)
3 � 4
(3, �1)
Chapter 4 Assessment Answer Key Vocabulary Test/Review Quiz (Lessons 4–1 and 4–2) Quiz (Lessons 4–5 and 4–6)
Page 230 Page 231 Page 232
An
swer
s
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© Glencoe/McGraw-Hill A34 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
1
impossible
Sample answer:a � b � c � 37
b � a � 4b � c � 29
yes
(1, 2)
y
xO
D � all real numbers;{R � y � y � �4}
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5.9
5050
impossible
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D
B
A
A
C
Chapter 4 Assessment Answer Key Mid-Chapter Test Cumulative ReviewPage 233 Page 234
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8 �16 �24
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1 3
© Glencoe/McGraw-Hill A35 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. 12.
13. 14.
15.
16.
17.
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.
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.
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.
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Chapter 1 Assessment Answer KeyStandardized Test Practice
Page 235 Page 236
An
swer
s
© Glencoe/McGraw-Hill A36 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25. (3, �4)
(2, �3)
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D�(5, 3); E�(4, �8);F�(�2, 1)
16 bedroom sets,5 living room sets
y
xO
4
8
12
4 8 12 16
(6, 10)
(16, 5)
(6, 5)
b � 6; � � 521b � 42� 546
(2, 1)
(�1, �3)
inconsistent
�3
Sample answer using (2, 16) and (3, 10):n � �6p � 28; 1
n
p
68
0
1012141618
1 2 3 4Price (dollars)
Num
ber
Sol
d
y � �2x � 7
y � �45
�x � �153�
19
�4�5 �3 �2 �1 0 2 3 4 5 61
all real numbers
0 1�3 �2 �1�4
�x � x �3 or x ��12
��
g � the number of additional games to
be won; �g
2�0
9� � 0.75;
at least 6 games
3
28
N, W, Z, Q, R
�19
Chapter 4 Assessment Answer Key Unit 1 TestPage 237 Page 238
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