chapter 4 resource masters - math class · 2019-11-24 · ©glencoe/mcgraw-hill iv glencoe algebra...

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








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

Matrix a 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 4439 2745 1692 5378 65

15 5 27 "423 6 0 514 70 24 "363 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


Equations Involving Matrices

Equal Matrices Two 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 equation4x # "2(x " 8) $ 2 Substitute x " 8 for y.4x # "2x $ 16 $ 2 Distributive Property6x # 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 equationy # 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 " yx $ y

17"8

2x $ 3y"4x $ 0.5y

7.58.0

3x $ 1.52y " 2.4

1$2

3$4

5$4

951

%3x

% $ %7y

% %2

x% $ 2y

"3"11

8x " 6y12x $ 4y

18 2012 "13

8x " y 16x 12 y " 4x

"1"18

5x $ 3y 2x " y

312

2x $ 3y x " 2y

"1 22

x " 2y3x " 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


Less

on

4-1


1. 2 ! 3 2. [0 15] 1 ! 2

3. 2 ! 2 4. 3 ! 3

5. 2 ! 4 6. 4 ! 1


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) 4yx " 3

x3y

6y x

2xy $ 2

4x $ 1 13 4z

5x4y " 3 8z

94y 9

x163z

7 2y " 412 28

3x $ 1 18 12 4z

3xy " 3

4x " 19y $ 5

5x $ 23y " 10

3x $ 27y " 2

"20 8y

5x24

10x 32

2056 " 6y

42

8 " x2y " 8

"14 2y

7x14

"1"1"1"3

9 3 "3 "63 4 "4 5

6 1 2"3 4 5"2 7 9

3 21 8

3 2 4"1 4 0



1. ["3 "3 7] 1 ! 3 2. 2 ! 3 3. 3 ! 4


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 32⎤⎢32 40 24⎥⎣24 32 24⎦

⎡40 32 24⎤⎢40 40 32⎥⎣32 24 24⎦

Week 1 Week 2 Week 3

Load 1 40 tons Load 1 40 tons Load 1 32 tons



⎡6 12 18⎤⎣8 15 22⎦

Child Student Adult

Cost Purchasedin Advance $6 $12 $18

Cost Purchasedat the Door $8 $15 $22

0"2

2x $ y3x $ 2y

"1 y

5x $ 8y3x " 11

y $ 8 17

3x y $ 4

"5 x " 1 3y 5z " 4

"5 3x $ 12y " 1 3z " 2

"3x " 21 8y " 5

6x " 12"3y $ 6

"3x"2y $ 16

"4x " 3 6y

202y $ 3

7x " 88y " 3

"36 17

6x2y $ 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


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

3xx $ yy " z

000

3 2 5"1 0 6


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


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. "

⎡ $14$ #$1

45$

⎤⎢ ⎥⎣#$

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 " 110 " ("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 $ 22 $ ("5) "12 $ ("6)

4 2"5 "6

6 "72 "12

⎡ 4 2⎤⎣#5 #6⎦

⎡6 #7⎤⎣2 #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 "51 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 "42 "5 "1

1 "2 5"3 4 1

⎡22 #15⎤⎣10 31⎦

2 14 3

6 "10"5 8

⎡#14 2⎤⎣ 8 6⎦

"2 0 2 5

3 "10 7

⎡#10 11⎤⎣ 12 #1⎦

"1 2"3 5

"4 5 2 3

⎡ 5 #2 #9⎤⎢ 1 11 #6⎥⎣12 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 "3060 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 " 021 " ("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



Operations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

ExampleExample

ExercisesExercises

Skills PracticeOperations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2


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 10⎤⎣14 5⎦

⎡ 15 #14⎤⎣#8 #1⎦

⎡ 15 #20⎤⎣#15 #5⎦

⎡6 6⎤⎣3 #6⎦

⎡2 8⎤⎣8 2⎦

⎡#1 0⎤⎣#3 #5⎦

⎡ 5 #2⎤⎣#2 #3⎦

⎡5 4⎤⎣5 1⎦

⎡#3 4⎤⎣ 3 1⎦

⎡2 2⎤⎣1 #2⎦

⎡3 2⎤⎣4 3⎦

⎡ 7 5 #1⎤⎣#24 9 21⎦

1 "1 56 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⎦

2 210

8 0"3

⎡ 5 #9⎤⎣#9 #17⎦

1 11 1

"2 5 5 9

⎡14 8 4⎤⎣ 5 14 #2⎦

9 9 24 6 4

5 "1 21 8 "6

4"1 2

⎡ 8 10⎤⎣#7 #3⎦

0 "76 2

8 3"1 "1



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 1,431,000⎤⎢73 1,574,000⎥⎣63 1,339,000⎦

⎡36 864,000⎤⎢32 672,000⎥⎣28 562,000⎦

⎡27 567,000⎤⎢41 902,000⎥⎣35 777,000⎦

Women Men

Businesses Loan Businesses Loan 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 3⎤⎣24 3⎦

27 "954 "18

2%3

8 12"16 20

3%4

⎡#12⎤⎣ #2⎦

1018

05

12

"1 4 "3 7 2 "6

2 "1 84 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 3 714 "9

Practice (Average)

Operations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

⎡16 #15 62⎤⎣14 45 75⎦

Reading to Learn MathematicsOperations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2

Pre-Activity How can matrices be used to calculate daily dietary needs?


• 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 falsec. M " (N " P) # (M " N) " P false d. k(M " N) # kM " kN true


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


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


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 possible⎡10 #16⎤⎢18 #6⎥⎣ 5 9⎦

⎡11 #21⎤⎣13 #15⎦

2 "2 3 1"2 4

2 0 "3 1 4 "2"1 3 1

1 "3"2 0

7 "25 "4

6 10"4 3"2 7

⎡24 #15⎤⎣14 #17⎦

⎡#1 4⎤⎢ 8 4⎥⎣#3 #9⎦

⎡#15 1 7⎤⎣ 26 #2 #12⎦

4 "1"2 5

5 "22 "3

1 22 1

3 "2 0 4"5 1

4 0 "2"3 1 1

"3 1 5 "2

⎡ 7 #7⎤⎣14 14⎦

⎡#3 #2⎤⎣ 2 34⎦

⎡ 12 3⎤⎣#6 9⎦

3 "12 4

3 "12 4

3 2"1 4

"1 0 3 7

3 00 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


Multiplicative Properties The Commutative Property of Multiplication does not holdfor matrices.

Properties of Matrix Multiplication For 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 4⎤⎣2 1⎦

⎡3 2⎤⎣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 "26 3

2 05 "3

1 "26 3

4 "32 1

"12 "21 "5 "20

6(1) $ ("3)(6) 6("2) $ ("3)(3)7(1) $ ("2)(6) 7("2) $ ("2)(3)

1 "26 3

6 "37 "2

1 "26 3

2 05 "3

4 "32 1

⎡1 #2⎤⎣6 3⎦

⎡2 0⎤⎣5 #3⎦

⎡4 #3⎤⎣2 1⎦


Multiplying Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3

ExampleExample

ExercisesExercises

Skills PracticeMultiplying Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3


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


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 #1⎤⎣1 0⎦

⎡#3 2⎤⎣ 5 1⎦

⎡2 1⎤⎣2 1⎦

⎡4⎤⎣4⎦

222

0 1 11 1 0

⎡#12 20⎤⎢ #6 8⎥⎣ 6 0⎦

3 "30 2

"4 4"2 1 2 3

⎡#6 6⎤⎢ 15 12⎥⎣ 3 #9⎦

0 33 0

2 "2 4 5"3 1

48

5 6"3

⎡#2 3 2⎤⎣ 6 #9 #6⎦

"1 3

0 "12 2

1 3"1 1

3"2

⎡#3⎤⎣#5⎦

3"2

1 3"1 1

⎡28 #19⎤⎣ 7 #9⎦

2 "53 1

5 62 1

21



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


7. & 8. &


11. [4 0 2] & [2] 12. & [4 0 2]

13. & 14. ["15 "9] & [#297 #75]


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

⎡172,452⎤⎢227,960⎥⎣125,642⎦

⎡$1796⎤⎢$2165⎥⎣$2538⎦

⎡36 24 22⎤⎢29 32 42⎥⎣18 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 3⎤⎣3 1⎦

6 1123 "10

⎡#30 10⎤⎣ 15 #5⎦

5 00 5

"6 2 3 "1

⎡ 4 0 2⎤⎢ 12 0 6⎥⎣#4 0 #2⎦

1 3"1

1 3"1

3 "2 76 0 "5

3 "2 76 0 "5

⎡#6 #12⎤⎣ 39 3⎦

2 47 "1

"3 0 2 5

⎡ 2 20⎤⎣#23 #5⎦

"3 0 2 5

2 47 "1

⎡30 #4 #6⎤⎣ 3 #6 26⎦

3 "2 76 0 "5

2 43 "1

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3

Reading to Learn MathematicsMultiplying Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3


Less

on

4-3

Pre-Activity How can matrices be used in sports statistics?


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.)

&


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⎦

⎡128 101⎤⎢205 166⎥⎣ 97 73⎦

Store Computers Printers

A 128 101

B 205 166

C 97 73

undefined3 ! 41 ! 14 ! 4

undefinedundefined2 ! 23 ! 3


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


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 30 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


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 the 90° 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 #1⎤⎣1 0⎦

⎡#2 1 0⎤⎣ 5 4 #1⎦

5 4 "13 "2 1

3 "2 15 4 "1

0 11 0

0 1"1 0

"1 0 0 "1

0 "11 0

0 11 0

"1 0 0 1

1 00 "1


Transformations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-44-4

ExampleExample

ExercisesExercises

Skills PracticeTransformations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-44-4


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.


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 0⎤⎣0 #1⎦

x

y

O

⎡4 0 2⎤⎣0 #1 3⎦

⎡#3 2 1⎤⎣ 1 #1 3⎦

x

y

O

⎡ 3 3 3⎤⎣#1 #1 #1⎦

x

y

O


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.


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 #2⎤⎣2 #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


Less

on

4-4

Pre-Activity How are transformations used in computer animation?


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


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⎦

⎡x⎤⎣y⎦

0 11 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


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 "73 4

"3 8 5 5

4 5"3 8

6 22 5

4 "32 5

5 5"3 "3

12 "3 5 3

4 "15 3

a b0 0

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-44-4

Study Guide and InterventionDeterminants

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5


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


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 2.13.4 5.3

20 1100.1 1.4

8.6 0.5 14 5

0.2 8"1.5 15

13 62"4 19

24 "8 7 "3

10 1622 40

"8 35 5 20

15 1223 28

14 8 9 "3

4 75 9

"6 "3"4 "1

5 12"7 "4

3 94 6

"8 3"2 1

6 "25 7

11 "5 9 3

⎢11 #5⎥⎢ 9 3⎥

6 3"8 5

⎢ 6 3⎥⎢#8 5⎥

a bc d

a b c d

ExampleExample

ExercisesExercises


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 23 0 "22 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 32 "3

1 02 6

3 0"3 6

4 5 "21 3 02 "3 6

⎢4 5 #2⎥⎢1 3 0⎥⎢2 #3 6⎥

a b 1 c d 1 e f 1

1%2

d eg h

d f g i

e f h i

a b c d e f g h i


Determinants

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5

ExampleExample

ExercisesExercises

Skills PracticeDeterminants

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5


Less

on

4-5


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. 403 2 21 "1 43 "1 0

3 "1 21 0 43 "2 0

2 "1 63 2 52 3 1

2 6 13 5 "12 1 "2

6 "1 15 2 "11 3 "2

2 "1 13 2 "12 3 "2

"1 6 2 5

2 2"1 4

"1 2 0 4

"1 "14 5 2

"1 "3 5 "2

3 "56 "11

"12 4 1 4

1 "3"3 4

1 "51 6

9 "2"4 1

"3 1 8 "7

"5 2 8 "6

3 122 8

0 95 8

2 53 1

1 61 7

10 9 5 8

5 21 3



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 31 8 00 5 "1

2 "1 "24 0 "20 3 2

1 2 "41 4 "62 3 3

1 "4 "11 "6 "22 3 1

2 2 31 "1 13 1 1

"4 3 "1 2 1 "2 4 1 "4

"12 0 3 7 5 "1 4 2 "6

2 7 "68 4 01 "1 3

0 "4 02 "1 13 "2 5

2 1 11 "1 "21 1 "1

2 "4 1 3 0 9"1 5 7

"2 3 1 0 4 "3 2 5 "1

0.5 "0.70.4 "0.3

2 "13 "9.5

3 "43.75 5

"1 "11 10 "2

3 "47 9

4 0"2 9

2 "55 "11

4 "3"12 4

"14 "3 2 "2

4 1"2 "5

9 63 2

1 62 7

Practice (Average)

Determinants

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5

Reading to Learn MathematicsDeterminants

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5


Less

on

4-5

Pre-Activity How are determinants used to find areas of polygons?


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. falseb. If both rows of a 2 ! 2 matrix are identical, the determinant of the matrix will be 0. truec. Every element of a 3 ! 3 matrix has a minor. trued. In order to evaluate a third-order determinant by expansion by minors it is necessary

to find the minor of every element of the matrix. falsee. If you evaluate a third-order determinant by expansion about the second row, the

position 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.


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


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

15 912

4 2 1218 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


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 # "7

5x " 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 bc d

a ec f $a bc d

e bf d $a bc d

ExampleExample

ExercisesExercises


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 52 3 "28 "2 10$$6 4 12 3 "28 "2 2

6 5 12 "2 "28 10 2$$6 4 12 3 "28 "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 cd e f g h i

a j cd k f g l i $a b cd e f g h i

j b ck e f l h i $a b cd e f g h i


Cramer’s Rule

NAME ______________________________________________ DATE ____________ PERIOD _____

4-64-6

ExampleExample

ExercisesExercises

Skills PracticeCramer’s Rule

NAME ______________________________________________ DATE ____________ PERIOD _____

4-64-6


Less

on

4-6


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)


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)



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)


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


Less

on

4-6

Pre-Activity How is Cramer’s Rule used to solve systems of equations?


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.


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 72 22$$3 22 "3

7 222 "3$$ 3 2 2 "3

3 2 2 "3$$ 7 222 "3

1$2

1$2

1%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 0⎤⎢0 1 0 0 0⎥⎢1 1 1 0 0⎥⎢1 1 0 1 0⎥⎣3 1 1 1 0⎦

⎡1 0 1 0 1 0⎤⎢0 1 0 0 1 1⎥⎢1 0 2 1 0 0⎥⎢0 1 1 2 0 0⎥⎢1 1 0 0 2 1⎥⎣0 1 0 1 0 0⎦

⎡3 1 2 0⎤⎢1 2 1 1⎥⎢1 1 2 1⎥⎣1 2 1 1⎦

⎡0 1 0 0 0⎤⎢1 0 0 0 0⎥⎢1 0 0 1 0⎥⎢1 0 1 0 0⎥⎣1 1 1 1 0⎦

⎡0 1 0 1 0 0⎤⎢0 0 1 0 0 0⎥⎢0 1 0 0 1 1⎥⎢1 0 0 0 1 0⎥⎢0 0 1 1 0 0⎥⎣1 0 0 0 0 0⎦

⎡0 1 1 1⎤⎢1 0 1 0⎥⎢1 1 0 0⎥⎣1 0 1 0⎦

25

1

3 4

26

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 j1 1 1 1 2

from1 2 1 1 1

computer i 1 2 1 1 11 2 0 4 11 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 j0 1 0 1 0

from0 0 0 1 1

computer i 1 0 0 1 01 1 1 0 10 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


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


yes no yes


no yes no

10. and 11. and 12. and

no yes yes

"5 3 7 "4

4 37 5

5 "2"17 7

7 217 5

3 2"4 "3

3 24 "6

%72% "%

32%

1 "23 72 4

"3 4 2 "%

52%

5 84 6

"%15% "%1

10%

%1

30% %1

10%

4 26 "2

"2 1 %

121% "%

52%

5 211 4

1 23 8

4 "15 3

"4 11 3 "8

8 113 4

2 3"1 "2

2 35 "1

2 "1 "%

52% %

32%

3 25 4

4 "5"3 4

4 53 4

1 00 1

21 " 20 12 " 12"35 $ 35 "20 $ 21

7 410 6

3 "2 "5 %

72%

1 00 1

21 " 20 "14 $ 1430 " 30 "20 $ 21

3 "2 "5 %

72%

7 410 6

⎡ 3 #2 ⎤⎢ ⎥⎣#5 $

72$ ⎦

⎡ 7 4⎤⎣10 6⎦

ExampleExample

ExercisesExercises


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⎦

8 210 4

3 64 8

6 510 8

⎡30 10⎤⎣20 40⎦

1$1000

⎡1 #1⎤⎣0 1⎦

40 "10"20 30

1 10 1

24 12 8 4

1 00 1

%73% " %

43% %

23% " %

23%

"%

134% $ %

134% "%

43% $ %

73%

7 22 1

%13% "%

23%

"%

23% %

73%

1 00 1

%73% " %

43% "%

134% $ %

134%

%

23% " %

23% "%

43% $ %

73%

%13% "%

23%

"%

23% %

73%

7 22 1

%13% "%

23%

"%

23% %

73%

1 "2"2 7

1%3

d "b"c a

1%ad " bc

7 22 1

⎡7 2⎤⎣2 1⎦

d "b"c a

1%ad " bc

a b c d


Identity and Inverse Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7

ExampleExample

ExercisesExercises

Skills PracticeIdentity and Inverse Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7


Less

on

4-7


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


9. # 10. #

11. no inverse exists 12.

13. 14. no inverse exists

15. 16. #

17. # 18. no inverse exists

19. # 20. ⎡2 0⎤⎣0 2⎦

1$4

2 00 2

⎡ #8 #8⎤⎣#10 10⎦

1$160

10 810 "8

10 8 5 4

⎡0 7⎤⎣7 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 "13 3

⎡ 0 4⎤⎣#6 #2⎦

1$24

"2 "4 6 0

9 36 2

⎡ 2 #1⎤⎣#3 1⎦

1 13 2

⎡ 0 #2⎤⎣#4 0⎦

1$8

0 24 0

"0.125 "0.125"0.125 "0.125

"4 "4"4 4

%121% %1

31%

"%1

11% %1

41%

4 "31 2

"%13% %

23%

%

16% %

16%

"1 4 1 2

0 "%17%

%

17% 0

0 7"7 0

2 "51 "2

"2 5"1 2

"1 0 0 "3

"1 0 0 3

"1 3 1 "2

2 31 1

1 0"1 1

1 01 1



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. false6. All square matrices have multiplicative identities. true


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 22 1

2$3

⎡1.5 6 7.5 3⎤⎣ 3 6 1.5 #1.5⎦

1.5 0 0 1.5

⎡1 4 5 2⎤⎣2 4 1 #1⎦

x

y

O(2, –1)

(1, 2)

(4, 4)

(5, 1)

4 66 9⎡ 1 5⎤

⎣#3 2⎦1

$172 "53 1

⎡3 #5⎤⎣1 2⎦

1$11

2 5"1 3⎡#7 #3⎤

⎣#4 #1⎦1$5

"1 3 4 "7

⎡ 5 0⎤⎣#3 2⎦

1$10

2 03 5⎡#3 #5⎤

⎣ 4 4⎦1$8

4 5"4 "3

%134% %

17%

%

17% %

37%

6 "2"2 3

%15% "%1

10%

%

25% %1

30%

3 1"4 2

3 25 3

"3 2 5 "3

"2 1 3 "2

2 13 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


Less

on

4-7

Pre-Activity How are inverse matrices used in cryptography?


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. falseb. There is a 3 ! 2 identity matrix. falsec. Two matrices are inverses of each other if their product is the identity matrix. trued. If M is a matrix, M"1 represents the reciprocal of M. falsee. No 3 ! 2 matrix has an inverse. truef. Every square matrix has an inverse. falseg. If the two columns of a 2 ! 2 matrix are identical, the matrix does not have an

inverse. 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.


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.


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 # 01 times a letter # the same letter0 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 0⎤⎢0 0 1⎥

⎡ H E S ⎤⎢ A W S ⎥⎣ I M H ⎦

⎡0 1 0⎤⎢0 0 1⎥⎣1 0 0⎦

⎡0 & 0 % 0 & 0 % 1 & 1 0 & 1 % 0 & 0 % 1 & 0 0 & 0 % 0 & 1 % 1 & 0⎤⎢1 & 0 % 0 & 0 % 0 & 1 1 & 1 % 0 & 0 % 0 & 0 1 & 0 % 0 & 1 % 0 & 0⎥⎣0 & 0 % 1 & 0 % 0 & 1 0 & 1 % 1 & 0 % 0 & 0 0 & 0 % 1 & 1 % 0 & 0⎦

⎡0 1 0⎤⎢0 0 1⎥⎣1 0 0⎦

0 0 11 0 00 1 0

⎡1 0⎤⎣0 1⎦

⎡0 & 1 % 1 & 1 0 & 1 % 1 & 0⎤⎣1 & 0 % 0 & 1 1 & 1 % 0 & 0⎦

⎡0 1⎤⎣1 0⎦

⎡0 1⎤⎣1 0⎦

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7

P # P"1 #

0 0 0 10 1 0 01 0 0 00 0 1 0

0 0 1 00 1 0 00 0 0 11 0 0 0

M #⎡ H E S ⎤⎢ A W S ⎥⎣ I M H ⎦

S H E S A WH I M

Study Guide and InterventionUsing Matrices to Solve Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-84-8


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

xy

3 "71 5

b. 2x " y $ 3z # "7x $ 3y " 4z # 157x $ 2y $ z # "28

& # "7 15"28

xy z

2 "1 3 1 3 "4 7 2 1

ExampleExample

ExercisesExercises


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⎤⎢ 60⎥⎣120⎦

⎡x⎤⎢y⎥⎣z⎦

⎡3 #2 1⎤⎢2 3 #1⎥⎣1 #4 2⎦

⎡#108⎤⎢ 40⎥⎣ 89⎦

⎡x⎤⎢y⎥⎣z⎦

⎡2 3 #9⎤⎢1 #2 5⎥⎣3 5 #4⎦

⎡27⎤⎢48⎥⎣72⎦

⎡x⎤⎢y⎥⎣z⎦

⎡1 #3 7⎤⎢2 1 #5⎥⎣4 #2 3⎦

⎡#100⎤⎢ #64⎥⎣ 8⎦

⎡x⎤⎢y⎥⎣z⎦

⎡4 #3 #1⎤⎢2 1 #3⎥⎣5 3 #2⎦

⎡ 32⎤⎢#18⎥⎣ 12⎦

⎡x⎤⎢y⎥⎣z⎦

⎡5 #1 7⎤⎢1 3 #2⎥⎣2 4 #3⎦

⎡12⎤⎢15⎥⎣25⎦

⎡x⎤⎢y⎥⎣z⎦

⎡2 1 7⎤⎢5 #1 3⎥⎣1 2 #6⎦

⎡24⎤⎣80⎦

⎡x⎤⎣y⎦

⎡3 #1⎤⎣2 3⎦

⎡18⎤⎣25⎦

⎡x⎤⎣y⎦

⎡5 2⎤⎣1 4⎦

⎡ 20⎤⎣#8⎦

⎡x⎤⎣y⎦

⎡4 #6⎤⎣3 1⎦

⎡ 15⎤⎣#10⎦

⎡x⎤⎣y⎦

⎡7 #2⎤⎣3 1⎦

⎡18⎤⎣12⎦

⎡x⎤⎣y⎦

⎡4 #3⎤⎣1 2⎦

⎡ 8⎤⎣#12⎦

⎡x⎤⎣y⎦

⎡2 1⎤⎣5 #3⎦


Solve Systems of Equations Use inverse matrices to solve systems of equationswritten as matrix equations.

Solving Matrix Equations If 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

xy

1 23 "1

"15 6

xy

3 65 9

4"8

xy

2 "32 5

3"7

xy

3 25 4

1612

xy

"4 "8 6 12

"2 18

xy

2 43 "1

2"2

xy

16"16

1%8

xy

1 00 1

64

4 "2"6 5

1%8

xy

5 26 4

4 "2"6 5

1%8

4 "2"6 5

1%8

4 "2"6 5

1%20 " 12

64

xy

5 26 4

⎡6⎤⎣4⎦

⎡ x⎤⎣ y⎦

⎡5 2⎤⎣6 4⎦


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


Less

on

4-8


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

& " & "


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

mn

5 612 "6

1$3

2512

cd

3 122 "6

1523

mn

7 "35 4

"1 7

ab

5 83 1

6"3

xy

4 31 3

"7"1

w z

1 34 3

⎡5⎤⎢0⎥⎣8⎦

⎡a⎤⎢b⎥⎣c⎦

⎡1 #1 1⎤⎢3 2 #1⎥⎣2 3 0⎦

⎡#4⎤⎢ 10⎥⎣ 3⎦

⎡x⎤⎢y⎥⎣z⎦

⎡ 6 #1 2⎤⎢#3 2 #1⎥⎣ 1 1 1⎦

⎡ 5⎤⎣10⎦

⎡x⎤⎣y⎦

⎡1 1⎤⎣3 2⎦

⎡ 1⎤⎣12⎦

⎡ r⎤⎣s⎦

⎡1 #1⎤⎣2 3⎦

⎡6⎤⎣7⎦

⎡c⎤⎣d⎦

⎡2 3⎤⎣3 #4⎦

⎡#3⎤⎣ 6⎦

⎡m⎤⎣ n⎦

⎡1 3⎤⎣4 3⎦

⎡16⎤⎣ 3⎦

⎡a⎤⎣b⎦

⎡3 8⎤⎣4 3⎦

⎡5⎤⎣1⎦

⎡x⎤⎣y⎦

⎡1 1⎤⎣2 #1⎦



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

& " & "


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

8 312 6

1$2

17"26

r s

"4 2 7 4

1634

cd

"5 3 6 4

12"11

ab

"1 "3 3 4

"7 10

xy

"2 3 1 5

0"2

gh

2 13 2

⎡#5⎤⎢ 8⎥⎣ 17⎦

⎡m⎤⎢ n ⎥⎣ p ⎦

⎡2 1 #3⎤⎢5 2 #2⎥⎣3 #3 5⎦

⎡ 3⎤⎢ 2⎥⎣#5⎦

⎡x⎤⎢y⎥⎣z⎦

⎡ 3 #2 5⎤⎢ 1 1 #4⎥⎣#2 2 7⎦

⎡10⎤⎣ 7⎦

⎡ r⎤⎣s⎦

⎡1 5⎤⎣2 #3⎦

⎡ 0⎤⎣#2⎦

⎡a⎤⎣b⎦

⎡2 1⎤⎣3 2⎦

⎡#2⎤⎣ 10⎦

⎡x⎤⎣y⎦

⎡6 #2⎤⎣3 3⎦

⎡ 9⎤⎣#13⎦

⎡x⎤⎣y⎦

⎡#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


Less

on

4-8

Pre-Activity How are inverse matrices used in population ecology?


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


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

xy z

3 2 "4 2 "1 0 0 5 6

⎡ 10⎤⎣#7⎦

⎡x⎤⎣y⎦

⎡3 5⎤⎣2 #4⎦

⎡140 500⎤⎣120 400⎦


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 61 2

1 "3"1 3

3 11 3

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-84-8

© Glencoe/McGraw-Hill A2 Glencoe Algebra 2

Answers (Lesson 4-1)

Stu

dy

Gu

ide

and I

nte

rven

tion

Intro

duct

ion

to M

atric

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-1

4-1

©G

lenc

oe/M

cGra

w-Hi

ll16

9G

lenc

oe A

lgeb

ra 2

Lesson 4-1

Org

aniz

e D

ata

Mat

rixa

rect

angu

lar a

rray

of v

aria

bles

or c

onst

ants

in h

orizo

ntal

rows

and

ver

tical

col

umns

, us

ually

enc

lose

d in

bra

cket

s.

A m

atri

x ca

n be

des

crib

ed b

y it

s d

imen

sion

s.A

mat

rix

wit

h m

row

s an

d n

colu

mns

is a

n m

!n

mat

rix.

Ow

ls’ e

ggs

incu

bate

for

30

day

s an

d t

hei

r fl

edgl

ing

per

iod

is

also

30

day

s.S

wif

ts’ e

ggs

incu

bate

for

20

day

s an

d t

hei

r fl

edgl

ing

per

iod

is

44 d

ays.

Pig

eon

egg

s in

cuba

te f

or 1

5 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

bate

for

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

tfinde

r

Wh

at a

re t

he

dim

ensi

ons

of m

atri

x A

if A

"?

Sinc

e m

atri

x A

has

2 ro

ws

and

4 co

lum

ns,t

he d

imen

sion

s of

Aar

e 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 a

gent

pro

vide

s fo

r po

tent

ial t

rave

lers

the

nor

mal

hig

h te

mpe

ratu

res

for

the

mon

ths

of J

anua

ry,A

pril,

July

,and

Oct

ober

for

var

ious

cit

ies.

In B

osto

n th

ese

figu

res

are

36°,

56°,

82°,

and

63°.

In D

alla

s th

ey a

re 5

4°,7

6°,9

7°,a

nd 7

9°.I

n L

os A

ngel

es t

hey

are

68°,

72°,

84°,

and

79°.

In S

eatt

le t

hey

are

46°,

58°,

74°,

and

60°,

and

in S

t.L

ouis

the

y ar

e38

°,67

°,89

°,an

d 69

°.O

rgan

ize

this

info

rmat

ion

in a

4 !

5 m

atri

x.So

urce

: The

New

York

Tim

es A

lman

ac

Bost

onDa

llas

Los

Ange

les

Seat

tleSt

.Lou

isJa

nuar

y⎡

3654

6846

38⎤

April

⎢56

7672

5867

⎥Ju

ly⎢

8297

8474

89⎥

Oct

ober

⎣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

⎦

Owl

Swift

Pige

onKi

ng P

engu

inIn

cuba

tion

3020

1553

Fl

edgl

ing

3044

1736

0

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-Hi

ll17

0G

lenc

oe A

lgeb

ra 2

Equ

atio

ns

Invo

lvin

g M

atri

ces

Equa

l Mat

rices

Two

mat

rices

are

equ

al if

they

hav

e th

e sa

me

dim

ensio

ns a

nd e

ach

elem

ent o

f one

mat

rix is

equa

l to

the

corre

spon

ding

ele

men

t of t

he o

ther

mat

rix.

You

can

use

the

defi

niti

on o

f eq

ual m

atri

ces

to s

olve

mat

rix

equa

tion

s.

Sol

ve

"

for

xan

d y

.

Sinc

e th

e m

atri

ces

are

equa

l,th

e co

rres

pond

ing

elem

ents

are

equ

al.W

hen

you

wri

te t

hese

nten

ces

to s

how

the

equ

alit

y,tw

o lin

ear

equa

tion

s ar

e fo

rmed

.4x

#"

2y$

2y

#x

"8

Thi

s sy

stem

can

be

solv

ed u

sing

sub

stit

utio

n.4x

#"

2y$

2Fi

rst e

quat

ion

4x#

"2(

x"

8) $

2Su

bstit

ute

x"

8 fo

r y.

4x#

"2x

$16

$2

Dist

ribut

ive P

rope

rty6x

#18

Add

2xto

eac

h sid

e.x

#3

Divid

e ea

ch s

ide

by 6

.

To fi

nd t

he v

alue

of y

,sub

stit

ute

3 fo

r x

in e

ithe

r eq

uati

on.

y#

x"

8Se

cond

equ

atio

ny

#3

"8

Subs

titut

e 3

for x

.y

#"

5Su

btra

ct.

The

sol

utio

n is

(3,

"5)

.

Sol

ve e

ach

equ

atio

n.

1.[5

x4y

] #[2

0 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 2

y"

2.4

1 $ 23 $ 4

5 $ 4

9

51

% 3x %

$% 7y %

% 2x %$

2y

"

3 "

11

8x

"6y

12x

$4y

18

20

12

"13

8

x"

y16

x

12y

"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⎦⎡4

x⎤⎣

y⎦

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Intro

duct

ion

to M

atric

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-1

4-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Intro

duct

ion

to M

atric

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-1

4-1

©G

lenc

oe/M

cGra

w-Hi

ll17

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

equ

atio

n.

7.[5

x3y

] #[1

512

](3

,4)

8.[3

x"

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

3y

6y

x

2x

y$

2

4x

$1

13

4z

5x

4y

"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

5x

24

10x

32

20

56

"6y

4

2

8

"x

2y

"8"

14

2y

7x

14

"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-Hi

ll17

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

equ

atio

n.

4.[4

x42

] #

[24

6y]

(6,7

)

5.["

2x22

"3z

] #[6

x"

2y45

](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 t

able

at

the

righ

t gi

ves

ti

cket

pri

ces

for

a co

ncer

t.W

rite

a 2

!3

mat

rix

that

rep

rese

nts

the

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.

Dur

ing

each

of

the

last

thr

ee w

eeks

,a

road

-bui

ldin

g cr

ew h

as u

sed

thre

e tr

uck-

load

s of

gra

vel.

The

tab

le a

t th

e ri

ght

show

s th

eam

ount

of

grav

el in

eac

h lo

ad.

15.W

rite

a m

atri

x fo

r th

e am

ount

of

grav

el

in e

ach

load

.

or

16.W

hat

are

the

dim

ensi

ons

of t

he 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

tons

Load

140

tons

Load

132

tons

Load

232

tons

Load

240

tons

Load

224

tons

Load

324

tons

Load

332

tons

Load

324

tons

⎡612

18⎤

⎣815

22⎦

Child

Stud

ent

Adul

t

Cost

Pur

chas

edin

Adv

ance

$6$1

2$1

8

Cost

Pur

chas

edat

the

Door

$8$1

5$2

2

0

"2

2x

$y

3x

$2y

"1

y

5x

$8y

3x

"11

y$

8

17

3x

y$

4"

5x

"1

3y

5z"

4

"5

3x$

1 2

y"

13z

"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 (A

vera

ge)

Intro

duct

ion

to M

atric

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-1

4-1



Rea

din

g t

o L

earn

Math

emati

csIn

trodu

ctio

n to

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-1

4-1

©G

lenc

oe/M

cGra

w-Hi

ll17

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

ucti

on t

o L

esso

n 4-

1 at

the

top

of

page

154

in y

our

text

book

.

Wha

t is

the

bas

e pr

ice

of a

Mid

-Siz

e 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.[1

40

"8

2]1

!5

2.Id

enti

fy e

ach

mat

rix

wit

h as

man

y of

the

fol

low

ing

desc

ript

ions

tha

t ap

ply:

row

mat

rix,

colu

mn

mat

rix,

squa

re m

atri

x,ze

ro m

atri

x.

a.[6

54

3]ro

w m

atrix

b.co

lum

n m

atrix

;zer

o m

atrix

c.[0

]ro

w m

atrix

;col

umn

mat

rix;s

quar

e m

atrix

;zer

o m

atrix

3.W

rite

a s

yste

m o

f eq

uati

ons

that

you

cou

ld u

se t

o so

lve

the

follo

win

g m

atri

x eq

uati

on f

orx,

y,an

d z.

(Do

not

actu

ally

sol

ve t

he s

yste

m.)

#3x

"#

9,x

%y

"5,

y#

z"

6

Hel

pin

g Y

ou

Rem

emb

er

4.So

me

stud

ents

hav

e tr

oubl

e re

mem

beri

ng w

hich

num

ber

com

es f

irst

in w

riti

ng t

hedi

men

sion

s of

a m

atri

x.T

hink

of

an e

asy

way

to

rem

embe

r th

is.

Sam

ple

answ

er:R

ead

the

mat

rix fr

om to

p to

bot

tom

,the

n fro

m le

ft to

right

.Rea

ding

dow

n gi

ves

the

num

ber o

f row

s,w

hich

is w

ritte

n fir

st in

the

dim

ensi

ons

of th

e m

atrix

.Rea

ding

acr

oss

give

s th

e nu

mbe

r of

colu

mns

,whi

ch is

writ

ten

seco

nd.

"9

5

6

3x

x$

y y

"z

0

0

0

3

25

"1

06

©G

lenc

oe/M

cGra

w-Hi

ll17

4G

lenc

oe A

lgeb

ra 2

Tess

ella

tions

A t

esse

llat

ion

is a

n ar

ange

men

t of

pol

ygon

s co

veri

ng a

pla

ne

wit

hout

any

gap

s or

ove

rlap

ping

.One

exa

mpl

e of

a t

esse

llati

on

is a

hon

eyco

mb.

Thr

ee c

ongr

uent

reg

ular

hex

agon

s m

eet

at

each

ver

tex,

and

ther

e is

no

was

ted

spac

e be

twee

n ce

lls.T

his

tess

ella

tion

is c

alle

d a

regu

lar

tess

ella

tion

sin

ce it

is f

orm

ed

by c

ongr

uent

reg

ular

pol

ygon

s.

A s

emi-

regu

lar

tess

ella

tion

is a

tes

sella

tion

for

med

by

two

or

mor

e re

gula

r po

lygo

ns s

uch

that

the

num

ber

of s

ides

of

the

poly

gons

mee

ting

at

each

ver

tex

is t

he s

ame.

For

exam

ple,

the

tess

ella

tion

at

the

left

has

tw

o re

gula

rdo

deca

gons

and

one

equ

ilate

ral t

rian

gle

mee

ting

at

each

vert

ex.W

e ca

n na

me

this

tes

sella

tion

a 3

-12-

12 f

or t

henu

mbe

r of

sid

es o

f ea

ch p

olyg

on t

hat

mee

t at

one

ver

tex.

Nam

e ea

ch s

emi-

regu

lar

tess

ella

tion

sh

own

acc

ord

ing

to t

he

nu

mbe

r of

sid

es o

f th

e p

olyg

ons

that

mee

t at

eac

h v

erte

x.

1.2.

3-3-

3-3-

64-

6-12

An

equ

ilat

eral

tri

angl

e,tw

o sq

uar

es,a

nd

a r

egu

lar

hex

agon

can

be

use

d t

o su

rrou

nd

a p

oin

t in

tw

o d

iffe

ren

t or

der

s.C

onti

nu

e ea

ch

pat

tern

to

see

wh

ich

is

a se

mi-

regu

lar

tess

ella

tion

.

3.3-

4-4-

64.

3-4-

6-4

not s

emi-r

egul

arse

mi-r

egul

ar

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

tion

.

5.3-

3-4-

126.

3-4-

3-12

7.4-

8-8

8.3-

3-3-

4-4

not s

emi-r

egul

arno

t sem

i-reg

ular

sem

i-reg

ular

sem

i-reg

ular

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-1

4-1


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Ope

ratio

ns w

ith M

atric

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-2

4-2

©G

lenc

oe/M

cGra

w-Hi

ll17

5G

lenc

oe A

lgeb

ra 2

Lesson 4-2

Ad

d a

nd

Su

btr

act

Mat

rice

s

Addi

tion

of M

atric

es$

#

Subt

ract

ion

of M

atric

es"

#

Fin

d A

%B

if A

"

and

B"

.

A$

B#

$

# #

Fin

d A

#B

if A

"

and

B"

.

A"

B#

"

#

#

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

poss

ible

.

1."

2.$

3.$

["6

3"

2]im

poss

ible

4.$

5."

6."

⎡$1 4$

#$ 14 5$

⎤⎢

⎥⎣#

$7 6$$1 61 $

⎦

⎡10

#1

#13

⎤⎢

19

#14

⎥⎣

1#

2#

2⎦

%1 2%%2 3%

%2 3%"

%1 2%

%3 4%

%2 5%

"%1 2%

%4 3%

"2

17

3

"4

3 "

85

6

8

0"

6

45

"11

"

73

4

⎡#6

4⎤⎢#

21⎥

⎣11

2⎦

"11

6

2"

5

4"

7

5

"2

"4

6

7

9

6

"3

2

⎡2#

211

⎤⎣3

131⎦

"4

32

6

9"

4

6"

59

"3

45

⎡12

4⎤⎣#

126⎦

"4

3

2"

12

8

7 "

10"

6

"6

11

5

"5

16

"1

"

2 "

48

"("

3)

3

"("

2)"

4 "

1 1

0 "

("6)

7 "

8

4

"3

"2

1

"6

8

"2

8

3

"4

10

7

⎡4

#3

⎤⎢ #

21

⎥⎣ #

68

⎦⎡#

28

⎤⎢

3#

4⎥

⎣ 10

7⎦

10

"5

"3

"18

6

$4

"7

$2

2 $

("5)

"12

$("

6)

4

2 "

5"

66

"7

2"

12

⎡4

2⎤⎣ #

5#

6⎦⎡6

#7⎤

⎣ 2#

12⎦a

"j

b"

kc

"l

d"

me

"n

f"o

g"

ph

"q

i"r

j

kl

mn

o

pq

r

ab

c

de

f g

hi

a$

jb

$k

c$

l

d$

me

$n

f$o

g$

ph

$q

i$r

j

kl

mn

o

pq

r

ab

c

de

f g

hi

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Scal

ar M

ult

iplic

atio

nYo

u ca

n m

ulti

ply

an m

!n

mat

rix

by a

sca

lar

k.

Scal

ar M

ultip

licat

ion

k#

If A

"

and

B"

,fin

d 3

B#

2A.

3B"

2A#

3"

2Su

bstit

utio

n

#

"M

ultip

ly.

#

"Si

mpl

ify.

#Su

btra

ct.

#Si

mpl

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

poss

ible

.

1.6

2."

3.0.

2

4.3

"2

5."

2$

4

6.2

$5

7.4

"2

8.8

$3

9.!

$"

⎡3

#1⎤

⎢⎥

⎣#$3 2$

$7 4$ ⎦

3"

5 1

7

91

"7

01 % 4

⎡28

8⎤⎢1

81⎥

⎣#7

20⎦

4

0 "

23

3

"4

2

1

3"

1 "

24

⎡#

4#

1428

⎤⎣#

1626

6⎦4

3"

4 2

"5

"1

1

"2

5 "

34

1⎡2

2#

15⎤

⎣10

31⎦

21

43

6

"10

"

58

⎡#14

2⎤⎣

86⎦

"2

0

25

3"

1 0

7⎡#

1011

⎤⎣

12#

1⎦"

12

"3

5"

45

2

3

⎡5

#2

#9⎤

⎢1

11#

6⎥⎣1

27

#19

⎦

⎡#

2#

5#

3⎤⎢#

1711

#8⎥

⎣6

#1

#15

⎦

⎡12

#30

18⎤

⎢0

42#

6⎥⎣#

2436

54⎦

25

"10

"45

555

"30

6

035

"95

6

159

51

"33

24

"18

345

1 % 3

2

"5

3

07

"1

"4

69

"11

15

33

18

"

3 "

815

"0

21

"("

12)

24 "

6

8

0 "

126

"3

15

21

24

2(

4)2(

0)

2("

6)2(

3)

3( "

1)3(

5)

3(

7)3(

8)

40

"6

3"

15

7

8

⎡#1

5⎤⎣

78⎦

⎡4

0⎤⎣ #

63⎦

ka

kbkc

k

dke

kf

ab

c

de

f

©G

lenc

oe/M

cGra

w-Hi

ll17

6G

lenc

oe A

lgeb

ra 2

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Ope

ratio

ns w

ith M

atric

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-2

4-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Ope

ratio

ns w

ith M

atric

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-2

4-2

©G

lenc

oe/M

cGra

w-Hi

ll17

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

poss

ible

.

1.[5

"4]

$[4

5][9

1]2.

"

3.[3

16]

$im

poss

ible

4.$

5.3[

94

"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"

,an

d C

"

to f

ind

th

e fo

llow

ing.

11.A

$B

12.B

"C

13.B

"A

14.A

$B

$C

15.3

B16

."5C

17.A

"4C

18.2

B$

3A⎡1

310

⎤⎣1

45⎦

⎡15

#14

⎤⎣#

8#

1⎦

⎡15

#20

⎤⎣#

15#

5⎦⎡6

6⎤⎣3

#6⎦

⎡28⎤

⎣82⎦

⎡#1

0⎤⎣#

3#

5⎦

⎡5

#2⎤

⎣#2

#3⎦

⎡54⎤

⎣51⎦

⎡#3

4⎤⎣

31⎦

⎡22⎤

⎣ 1#

2⎦⎡3

2⎤⎣ 4

3⎦

⎡7

5#

1⎤⎣#

249

21⎦

1"

15

66

"3

3

13

"4

75

⎡#8

40⎤

⎢44

1⎥⎣#

35⎦

6

5 "

3"

2

10

"4

6 1

01

"1

1

⎡16

⎤⎢

#8⎥

⎣#49

⎦

2

2

10

8

0

"3

⎡5

#9⎤

⎣#9

#17

⎦1

1 1

1"

25

5

9

⎡14

84⎤

⎣5

14#

2⎦9

92

46

4 5

"1

2 1

8"

6

4 "

1

2

⎡8

10⎤

⎣#7

#3⎦

0"

7 6

2

83

"1

"1

©G

lenc

oe/M

cGra

w-Hi

ll17

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

poss

ible

.

1.$

2.

"

3."

3$

44.

7"

2

5."

2$

4"

6.

$

Use

A"

,B"

,an

d C

"

to f

ind

th

e fo

llow

ing.

7.A

"B

8.A

"C

9."

3B10

.4B

"A

11."

2B"

3C12

.A$

0.5C

ECO

NO

MIC

SF

or E

xerc

ises

13

an

d 1

4,u

se t

he

tabl

e th

at

show

s lo

ans

by a

n e

con

omic

dev

elop

men

t bo

ard

to

wom

en

and

men

sta

rtin

g n

ew b

usi

nes

ses.

13.W

rite

tw

o m

atri

ces

that

re

pres

ent

the

num

ber

of n

ewbu

sine

sses

and

loan

am

ount

s,on

e fo

r w

omen

and

one

for

men

.

,

14.F

ind

the

sum

of

the

num

bers

of

new

bus

ines

ses

and

loan

am

ount

s fo

r bo

th m

en a

nd 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 t

able

tha

t gi

ves

nutr

itio

nal

info

rmat

ion

for

two

type

s of

dog

foo

d.F

ind

the

diff

eren

ce in

the

per

cent

of

prot

ein,

fat,

and

fibe

rbe

twee

n M

ix B

and

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

0⎤⎢7

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⎦

Wom

enM

en

Busi

ness

esLo

anBu

sine

sses

Loan

Am

ount

($)

Amou

nt ($

)

1999

27$5

67,0

0036

$864

,000

2000

41$9

02,0

0032

$672

,000

2001

35$7

77,0

0028

$562

,000

⎡9

#5

3⎤⎣#

64

12⎦

⎡#26

16#

28⎤

⎣16

12#

42⎦

⎡#12

1720

⎤⎣

7#

6#

38⎦

⎡6

#12

#15

⎤⎣#

30

27⎦

⎡#6

7#

6⎤⎣

310

#18

⎦⎡

6#

5#

5⎤⎣#

46

11⎦

⎡10

#8

6⎤⎣ #

6#

420

⎦⎡#

24

5⎤⎣

10

#9⎦

⎡4

#1

0⎤⎣ #

36

2⎦

⎡24

3⎤⎣2

43⎦

27

"9

54

"18

2 % 3

8

12

"16

20

3 % 4⎡#

12⎤

⎣#

2⎦1

0 1

80

5

1

2

"1

4"

3

72

"6

2"

18

47

9⎡

#9

64⎤

⎣#13

581

⎦

"3

16

"21

12

"1

0 1

7"

11

⎡71

⎤⎢#

116⎥

⎣42

⎦

"67

45

"24

4

"71

18

⎡#4

8⎤⎢1

0#

4⎥⎣

68⎦

"6

9

7"

11

"8

17

2

"1

3

7 1

4"

9Pra

ctic

e (A

vera

ge)

Ope

ratio

ns w

ith M

atric

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-2

4-2

⎡16

#15

62⎤

⎣14

4575

⎦


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csO

pera

tions

with

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-2

4-2

©G

lenc

oe/M

cGra

w-Hi

ll17

9G

lenc

oe A

lgeb

ra 2

Lesson 4-2

Pre-

Act

ivit

yH

ow c

an m

atri

ces

be u

sed

to

calc

ula

te d

aily

die

tary

nee

ds?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 4-

2 at

the

top

of

page

160

in y

our

text

book

.

•W

rite

a s

um t

hat

repr

esen

ts t

he t

otal

num

ber

of C

alor

ies

in t

he p

atie

nt’s

diet

for

Day

2.(

Do

not

actu

ally

cal

cula

te t

he s

um.)

482

%62

2 %

987

•W

rite

the

sum

tha

t re

pres

ents

the

tot

al f

at c

onte

nt in

the

pat

ient

’s d

iet

for

Day

3.(

Do

not

actu

ally

cal

cula

te t

he s

um.)

11 %

12 %

38

Rea

din

g t

he

Less

on

1.Fo

r ea

ch p

air

of m

atri

ces,

give

the

dim

ensi

ons

of t

he in

dica

ted

sum

,dif

fere

nce,

or s

cala

rpr

oduc

t.If

the

indi

cate

d su

m,d

iffer

ence

,or

scal

ar p

rodu

ct d

oes

not

exis

t,w

rite

impo

ssib

le.

A#

B#

C#

D#

A$

D:

C$

D:

5B:

"4C

:2D

"3A

:

2.Su

ppos

e th

at M

,N,a

nd P

are

nonz

ero

2 !

4 m

atri

ces

and

kis

a n

egat

ive

real

num

ber.

Indi

cate

whe

ther

eac

h of

the

fol

low

ing

stat

emen

ts is

tru

eor

fal

se.

a.M

$(N

$P

) #M

$(P

$N

) tru

eb.

M"

N#

N"

Mfa

lse

c.M

"(N

"P

) #(M

"N

) "P

fals

ed.

k(M

"N

) #kM

"kN

true

Hel

pin

g Y

ou

Rem

emb

er

3.T

he m

athe

mat

ical

ter

m s

cala

rm

ay b

e un

fam

iliar

,but

its

mea

ning

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

usag

e of

the

wor

d sc

ale

help

you

rem

embe

r th

e m

eani

ng o

f sca

lar?

Sam

ple

answ

er:A

sca

le o

f mile

s te

lls y

ou h

ow to

mul

tiply

the

dist

ance

syo

u m

easu

re o

n a

map

by

a ce

rtai

n nu

mbe

r to

get t

he a

ctua

l dis

tanc

ebe

twee

n tw

o lo

catio

ns.T

his

mul

tiplie

r is

ofte

n ca

lled

a sc

ale

fact

or.A

scal

arre

pres

ents

the

sam

e id

ea:I

t is

a re

al n

umbe

r by

whi

ch a

mat

rixca

n be

mul

tiplie

d to

cha

nge

all t

he e

lem

ents

of t

he m

atrix

by

a un

iform

scal

efa

ctor

.

2 !

33

!2

2 !

2im

poss

ible

2 !

3

"3

60

"8

40

5

10

"3

6

412

"4

0

0"

5

35

6 "

28

1

©G

lenc

oe/M

cGra

w-Hi

ll18

0G

lenc

oe A

lgeb

ra 2

Sund

aram

’s S

ieve

The

pro

pert

ies

and

patt

erns

of

prim

e nu

mbe

rs h

ave

fasc

inat

ed m

any

mat

hem

atic

ians

.In

193

4,a

youn

g E

ast

Indi

an s

tude

nt n

amed

Sun

dara

m c

onst

ruct

ed t

he f

ollo

win

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

urpr

isin

g pr

oper

ty o

f th

is m

atri

x is

tha

t it

can

be

used

to

dete

rmin

e w

heth

er o

r no

t so

me

num

bers

are

pri

me.

Com

ple

te t

hes

e p

robl

ems

to d

isco

ver

this

pro

per

ty.

1.T

he f

irst

row

and

the

fir

st c

olum

n ar

e cr

eate

d by

usi

ng a

n ar

ithm

etic

pa

tter

n.W

hat

is t

he c

omm

on d

iffe

renc

e us

ed in

the

pat

tern

?3

2.F

ind

the

next

fou

r nu

mbe

rs in

the

fir

st r

ow.

28,3

1,34

,37

3.W

hat

are

the

com

mon

dif

fere

nces

use

d to

cre

ate

the

patt

erns

in r

ows

2,3,

4,an

d 5?

5,7,

9,11

4.W

rite

the

nex

t tw

o ro

ws

of t

he m

atri

x.In

clud

e ei

ght

num

bers

in e

ach

row

.ro

w 6

:19,

32,4

5,58

,71,

84,9

7,11

0;ro

w 7

:22,

37,5

2,67

,82,

97,1

12,1

27

5.C

hoos

e an

y fi

ve n

umbe

rs f

rom

the

mat

rix.

For

each

num

ber

n,th

at y

ou

chos

e fr

om t

he m

atri

x,fi

nd 2

n$

1.An

swer

s w

ill v

ary.

6.W

rite

the

fac

tori

zati

on o

f ea

ch v

alue

of

2n$

1 th

at y

ou f

ound

in p

robl

em 5

.An

swer

s w

ill v

ary,

but a

ll nu

mbe

rs a

re c

ompo

site

.

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 in

the

mat

rix,

then

2n

$1

(is/

is n

ot)

a pr

ime

num

ber.

8.C

hoos

e an

y fi

ve n

umbe

rs t

hat

are

not

in t

he m

atri

x.F

ind

2n$

1 fo

r ea

ch

of t

hese

num

bers

.Sho

w t

hat

each

res

ult

is a

pri

me

num

ber.

Answ

ers

will

var

y,bu

t all

num

bers

are

prim

e.

9.C

ompl

ete

this

sta

tem

ent:

If n

does

not

occ

ur in

the

mat

rix,

then

2n

$1

is

.a

prim

e nu

mbe

r

is n

ot

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-2

4-2



Stu

dy

Gu

ide

and I

nte

rven

tion

Mul

tiply

ing

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-3

4-3

©G

lenc

oe/M

cGra

w-Hi

ll18

1G

lenc

oe A

lgeb

ra 2

Lesson 4-3

Mu

ltip

ly M

atri

ces

You

can

mul

tipl

y tw

o m

atri

ces

if a

nd o

nly

if t

he n

umbe

r of

col

umns

in t

he f

irst

mat

rix

is e

qual

to

the

num

ber

of r

ows

in t

he s

econ

d m

atri

x.

Mul

tiplic

atio

n of

Mat

rices

& #

Fin

d A

Bif

A"

and

B"

.

AB

#

& Su

bstit

utio

n

#M

ultip

ly co

lum

ns b

y ro

ws.

#Si

mpl

ify.

Fin

d e

ach

pro

du

ct,i

f p

ossi

ble.

1.&

2.&

3.&

4.&

5.&

6.&

7.&

[04

"3]

8.&

9.&

not p

ossi

ble

⎡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

7⎦a

1x1

$b 1

x 2a 1

y 1$

b 1y 2

a2x

1$

b 2x 2

a 2y 1

$b 2

y 2

x1

y 1

x2

y 2

a1

b 1

a2

b 2

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-Hi

ll18

2G

lenc

oe A

lgeb

ra 2

Mu

ltip

licat

ive

Pro

per

ties

The

Com

mut

ativ

e P

rope

rty

of M

ulti

plic

atio

n do

es n

otho

ldfo

r m

atri

ces.

Prop

ertie

s of

Mat

rix M

ultip

licat

ion

For a

ny m

atric

es A

, B, a

nd C

for w

hich

the

mat

rix p

rodu

ct is

defin

ed, a

nd a

ny s

cala

r c, t

he fo

llowi

ng p

rope

rties

are

true

.

Asso

ciat

ive

Prop

erty

of M

atrix

Mul

tiplic

atio

n(A

B)C

#A(

BC)

Asso

ciat

ive

Prop

erty

of S

cala

r Mul

tiplic

atio

nc(

AB) #

(cA)

B#

A(cB

)

Left

Dist

ribut

ive

Prop

erty

C(A

$B

) #C

A$

CB

Righ

t Dis

tribu

tive

Prop

erty

(A$

B)C

#AC

$BC

Use

A"

,B"

,an

d C

"

to f

ind

eac

h p

rod

uct

.

a.(A

$B

)C

(A$

B)C

#!

$"&

#

&

# #

b.A

C$

BC

AC

$B

C#

& $

&

#$

#$

#

Not

e th

at a

ltho

ugh

the

resu

lts

in t

he e

xam

ple

illus

trat

e th

e R

ight

Dis

trib

utiv

e P

rope

rty,

they

do

not

prov

e 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

) #(c

A)B

yes

2.A

B#

BA

no

3.B

C#

CB

no4.

(AB

)C#

A(B

C)

yes

5.C

(A$

B) #

AC

$B

Cno

6.c(

A$

B) #

cA$

cBye

s

⎡ #$1 2$

#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

3⎦⎡2

0⎤⎣ 5

#3⎦

⎡4#

3⎤⎣ 2

1⎦

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Mul

tiply

ing

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-3

4-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lencoe/McG

raw-HillA9

Glencoe Algebra 2

Answers

Answers

(Lesson 4-3)

Skills PracticeMultiplying Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3


Less

on

4-3


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


7. [3 2] & [8] 8. &


11. ["3 4] & [8 11] 12. & [2 "3 "2]

13. & not possible 14. &

15. & 16. &


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 #1⎤⎣1 0⎦

⎡#3 2⎤⎣ 5 1⎦

⎡2 1⎤⎣2 1⎦

⎡4⎤⎣4⎦

222

0 1 11 1 0

⎡#12 20⎤⎢ #6 8⎥⎣ 6 0⎦

3 "30 2

"4 4"2 1 2 3

⎡#6 6⎤⎢ 15 12⎥⎣ 3 #9⎦

0 33 0

2 "2 4 5"3 1

48

5 6"3

⎡#2 3 2⎤⎣ 6 #9 #6⎦

"1 3

0 "12 2

1 3"1 1

3"2

⎡#3⎤⎣#5⎦

3"2

1 3"1 1

⎡28 #19⎤⎣ 7 #9⎦

2 "53 1

5 62 1

21



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


7. & 8. &


11. [4 0 2] & [2] 12. & [4 0 2]

13. & 14. ["15 "9] & [#297 #75]


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

⎡172,452⎤⎢227,960⎥⎣125,642⎦

⎡$1796⎤⎢$2165⎥⎣$2538⎦

⎡36 24 22⎤⎢29 32 42⎥⎣18 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 3⎤⎣3 1⎦

6 1123 "10

⎡#30 10⎤⎣ 15 #5⎦

5 00 5

"6 2 3 "1

⎡ 4 0 2⎤⎢ 12 0 6⎥⎣#4 0 #2⎦

1 3"1

1 3"1

3 "2 76 0 "5

3 "2 76 0 "5

⎡#6 #12⎤⎣ 39 3⎦

2 47 "1

"3 0 2 5

⎡ 2 20⎤⎣#23 #5⎦

"3 0 2 5

2 47 "1

⎡30 #4 #6⎤⎣ 3 #6 26⎦

3 "2 76 0 "5

2 43 "1

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3



Rea

din

g t

o L

earn

Math

emati

csM

ultip

lyin

g M

atric

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-3

4-3

©G

lenc

oe/M

cGra

w-Hi

ll18

5G

lenc

oe A

lgeb

ra 2

Lesson 4-3

Pre-

Act

ivit

yH

ow c

an m

atri

ces

be u

sed

in

sp

orts

sta

tist

ics?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 4-

3 at

the

top

of

page

167

in y

our

text

book

.

Wri

te a

sum

tha

t sh

ows

the

tota

l poi

nts

scor

ed b

y th

e O

akla

nd R

aide

rsdu

ring

the

200

0 se

ason

.(T

he s

um w

ill in

clud

e m

ulti

plic

atio

ns.D

o no

tac

tual

ly c

alcu

late

thi

s su

m.)

6 &

58 %

1 &

56 %

3 &

23 %

2 &

1 %

2 &

2

Rea

din

g t

he

Less

on

1.D

eter

min

e w

heth

er e

ach

indi

cate

d m

atri

x pr

oduc

t is

def

ined

.If

so,s

tate

the

dim

ensi

ons

of t

he p

rodu

ct.I

f no

t,w

rite

und

efin

ed.

a.M

3 !

2an

d N

2 !

3M

N:

NM

:

b.M

1 !

2an

d N

1 !

2M

N:

NM

:

c.M

4 !

1an

d N

1 !

4M

N:

NM

:

d.M

3 !

4an

d N

4 !

4M

N:

NM

:

2.T

he r

egio

nal s

ales

man

ager

for

a c

hain

of

com

pute

r st

ores

wan

ts t

o co

mpa

re t

he r

even

uefr

om s

ales

of

one

mod

el o

f no

tebo

ok c

ompu

ter

and

one

mod

el o

f pr

inte

r fo

r th

ree

stor

esin

his

are

a.T

he n

oteb

ook

com

pute

r se

lls f

or $

1850

and

the

pri

nter

for

$17

5.T

he n

umbe

rof

com

pute

rs a

nd p

rint

ers

sold

at

the

thre

e st

ores

dur

ing

Sept

embe

r ar

e sh

own

in t

hefo

llow

ing

tabl

e.

Wri

te a

mat

rix

prod

uct

that

the

man

ager

cou

ld u

se t

o fi

nd t

he t

otal

rev

enue

for

com

pute

rs a

nd p

rint

ers

for

each

of

the

thre

e st

ores

.(D

o no

t ca

lcul

ate

the

prod

uct.

)

&

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stud

ents

fin

d th

e pr

oced

ure

of m

atri

x m

ulti

plic

atio

n co

nfus

ing

at f

irst

bec

ause

itis

unf

amili

ar.T

hink

of

an e

asy

way

to

use

the

lett

ers

R a

nd C

to

rem

embe

r ho

w t

om

ulti

ply

mat

rice

s an

d w

hat

the

dim

ensi

ons

of t

he p

rodu

ct w

ill b

e.Sa

mpl

e an

swer

:Ju

st re

mem

ber R

C fo

r “ro

w,co

lum

n.”M

ultip

ly e

ach

row

of th

e fir

st m

atrix

by e

ach

colu

mn

of th

e se

cond

mat

rix.T

he d

imen

sion

s of

the

prod

uct a

reth

e nu

mbe

r of r

ows

of th

e fir

st m

atrix

and

the

num

ber o

f col

umns

of th

ese

cond

mat

rix.

⎡185

0⎤⎣

175⎦

⎡128

101⎤

⎢205

166⎥

⎣97

73⎦

Stor

eCo

mpu

ters

Prin

ters

A12

810

1

B20

516

6

C97

73

unde

fined

3 !

41

!1

4 !

4un

defin

edun

defin

ed2

!2

3 !

3

©G

lenc

oe/M

cGra

w-Hi

ll18

6G

lenc

oe A

lgeb

ra 2

Four

th-O

rder

Det

erm

inan

tsTo

find

the

val

ue o

f a

4 !

4 de

term

inan

t,us

e a

met

hod

calle

d ex

pan

sion

by

min

ors.

Fir

st w

rite

the

exp

ansi

on.U

se t

he f

irst

row

of

the

dete

rmin

ant.

Rem

embe

r th

at t

he s

igns

of

the

term

s al

tern

ate.

###

##

##

##

#T

hen

eval

uate

eac

h 3

!3

dete

rmin

ant.

Use

any

row

.

##

##

##

##

##

#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

"4

60

#"

30

35

01

"4

6"

20

45

2"

4#

"("

2)4

35

21

"4

0"

20

04

30

21

60

"2

" 7

04

50

2"

46

00

$2

03

50

1"

46

"2

0"

("3)

43

52

1"

40

"2

0#

6

6"

32

70

43

50

21

"4

60

"2

0

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-3

4-3


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Tran

sfor

mat

ions

with

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-Hi

ll18

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 c

oord

inat

es o

f po

ints

on

a pl

ane

are

usef

ul in

des

crib

ing

tran

sfor

mat

ions

.

Tran

slat

ion

a tra

nsfo

rmat

ion

that

mov

es a

figu

re fr

om o

ne lo

catio

n to

ano

ther

on

the

coor

dina

te p

lane

You

can

use

mat

rix

addi

tion

and

a t

rans

lati

on m

atri

x to

fin

d th

e co

ordi

nate

s of

the

tran

slat

ed f

igur

e.

Dila

tion

a tra

nsfo

rmat

ion

in w

hich

a fi

gure

is e

nlar

ged

or re

duce

d

You

can

use

scal

ar m

ulti

plic

atio

n to

per

form

dila

tion

s.

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 v

erte

x m

atri

x fo

r !

AB

C.

Add

the

tra

nsla

tion

mat

rix

to t

he v

erte

x m

atri

x of

!A

BC

. $

#

The

coo

rdin

ates

of

the

vert

ices

of !

A'B

'C'

are

A'(

1,0)

,B'(

5,1)

,and

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

the

tra

nsla

tion

mat

rix.

2.F

ind

the

coor

dina

tes

of t

he v

erti

ces

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

the

coo

rdin

ates

of

the

vert

ices

of !

AB

Cin

a v

erte

x m

atri

x.

4.F

ind

the

coor

dina

tes

of t

he v

erti

ces

of im

age

!A

'B'C

'.

A'!1,

#",B

' !,2

",C' !2

,"

5.G

raph

the

pre

imag

e an

d th

e im

age.

1 $ 21 $ 2

1 $ 2⎡4

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-Hi

ll18

8G

lenc

oe A

lgeb

ra 2

Ref

lect

ion

s an

d R

ota

tio

ns

Refle

ctio

nFo

r a re

flect

ion

over

the:

x-ax

isy-

axis

line

y#

x

Mat

rices

mul

tiply

the

verte

x m

atrix

on

the

left

by:

Rota

tion

For a

cou

nter

clock

wise

rota

tion

abou

t the

90°

180°

270°

Mat

rices

orig

in o

f:

mul

tiply

the

verte

x m

atrix

on

the

left

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 o

rder

ed p

airs

as

a ve

rtex

mat

rix.

The

n m

ulti

ply

the

vert

ex m

atri

x by

the

refl

ecti

on m

atri

x fo

r y

#x.

& #

The

coo

rdin

ates

of

the

vert

ices

of A

'B'C

'ar

e A

'(5,

3),B

'(4,

"2)

,and

C'(

"1,

1).

1.T

he c

oord

inat

es o

f th

e ve

rtic

es o

f qu

adri

late

ral A

BC

Dar

e A

("2,

1),B

("1,

3),C

(2,2

),an

dD

(2,"

1).W

hat

are

the

coor

dina

tes

of t

he v

erti

ces

of t

he 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

vert

ices

D("

2,5)

,E(1

,4),

and

F(0

,"1)

is r

otat

ed 9

0°co

unte

rclo

ckw

ise

abou

t th

e or

igin

.

a.W

rite

the

coo

rdin

ates

of

the

tria

ngle

in a

ver

tex

mat

rix.

b.W

rite

the

rot

atio

n m

atri

x fo

r th

is s

itua

tion

.

c.F

ind

the

coor

dina

tes

of t

he v

erti

ces

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#

1⎤⎣1

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

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Tran

sfor

mat

ions

with

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-4

4-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Tran

sfor

mat

ions

with

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-Hi

ll18

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 ve

rtic

es A

(2,3

),B

(0,4

),an

d C

("3,

"3)

is

tran

slat

ed 3

uni

ts r

ight

and

1 u

nit

dow

n.

1.W

rite

the

tra

nsla

tion

mat

rix.

2.F

ind

the

coor

dina

tes

of !

A'B

'C'.

A'(5

,2),

B'(3

,3),

C'(0

,#4)

3.G

raph

the

pre

imag

e an

d th

e im

age.

For

Exe

rcis

es 4

–6,u

se t

he

foll

owin

g in

form

atio

n.

The

ver

tice

s of

!R

ST

are

R("

3,1)

,S(2

,"1)

,and

T(1

,3).

The

tr

iang

le is

dila

ted

so t

hat

its

peri

met

er is

tw

ice

the

orig

inal

pe

rim

eter

.

4.W

rite

the

coo

rdin

ates

of !

RS

Tin

a

vert

ex m

atri

x.

5.F

ind

the

coor

dina

tes

of t

he im

age

!R

'S'T

'.R

'(#6,

2),S

'(4,#

2),T

'(2,6

)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 v

erti

ces

of !

DE

Far

e D

(4,0

),E

(0,"

1),a

nd F

(2,3

).T

he t

rian

gle

is r

efle

cted

ove

r th

e x-

axis

.

7.W

rite

the

coo

rdin

ates

of !

DE

Fin

a

vert

ex m

atri

x.

8.W

rite

the

ref

lect

ion

mat

rix

for

this

sit

uati

on.

9.F

ind

the

coor

dina

tes

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

vert

ices

X(1

,"3)

,Y("

4,1)

,and

Z("

2,5)

is

rota

ted

180º

cou

nter

cloc

kwis

e ab

out

the

orig

in.

11.W

rite

the

coo

rdin

ates

of

the

tria

ngle

in a

ve

rtex

mat

rix.

12.W

rite

the

rot

atio

n m

atri

x fo

r th

is s

itua

tion

.

13.F

ind

the

coor

dina

tes

of !

X'Y

'Z'.

X'(#

1,3)

,Y'(4

,#1)

,Z'(2

,#5)

14.G

raph

the

pre

imag

e an

d th

e im

age.

⎡#1

0⎤⎣

0#

1⎦

⎡1

#4

#2⎤

⎣#3

15⎦

X'

Y'

Z'

X

Y

Z

x

y

O

⎡10⎤

⎣0#

1⎦D

'E'

F'

DE

F

x

y

O

⎡40

2⎤⎣0

#1

3⎦

⎡#3

21⎤

⎣1

#1

3⎦R'

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-Hi

ll19

0G

lenc

oe A

lgeb

ra 2

For

Exe

rcis

es 1

–3,u

se t

he

foll

owin

g in

form

atio

n.

Qua

drila

tera

l WX

YZ

wit

h ve

rtic

es W

("3,

2),X

("2,

4),Y

(4,1

),an

d Z

(3,0

) is

tra

nsla

ted

1 un

it le

ft a

nd 3

uni

ts d

own.

1.W

rite

the

tra

nsla

tion

mat

rix.

2.F

ind

the

coor

dina

tes

of q

uadr

ilate

ral W

'X'Y

'Z'.

W'(#

4,#

1),X

'(#3,

1),Y

'(3,#

2),Z

'(2,#

3)3.

Gra

ph t

he p

reim

age

and

the

imag

e.

For

Exe

rcis

es 4

–6,u

se t

he

foll

owin

g in

form

atio

n.

The

ver

tice

s of

!R

ST

are

R(6

,2),

S(3

,"3)

,and

T("

2,5)

.The

tri

angl

e is

dila

ted

so t

hat

its

peri

met

er is

one

hal

f th

e or

igin

al p

erim

eter

.

4.W

rite

the

coo

rdin

ates

of !

RS

Tin

a v

erte

x m

atri

x.

5.F

ind

the

coor

dina

tes

of t

he im

age

!R

'S'T

'.R

'(3,1

),S

'(1.5

,#1.

5),T

'(#1,

2.5)

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 v

erti

ces

of q

uadr

ilate

ral A

BC

Dar

e A

("3,

2),B

(0,3

),C

(4,"

4),

and

D("

2,"

2).T

he q

uadr

ilate

ral i

s re

flec

ted

over

the

y-a

xis.

7.W

rite

the

coo

rdin

ates

of A

BC

Din

a

vert

ex m

atri

x.

8.W

rite

the

ref

lect

ion

mat

rix

for

this

sit

uati

on.

9.F

ind

the

coor

dina

tes

of A

'B'C

'D'.

A'(3

,2),

B'(0

,3),

C'(#

4,#

4),D

'(2,#

2)10

.Gra

ph A

BC

Dan

d A

'B'C

'D'.

11.A

RC

HIT

ECTU

RE

Usi

ng a

rchi

tect

ural

des

ign

soft

war

e,th

e B

radl

eys

plot

the

ir k

itch

enpl

ans

on a

gri

d w

ith

each

uni

t re

pres

enti

ng 1

foo

t.T

hey

plac

e th

e co

rner

s of

an

isla

nd a

t(2

,8),

(8,1

1),(

3,5)

,and

(9,

8).I

f th

e B

radl

eys

wis

h to

mov

e th

e is

land

1.5

fee

t to

the

righ

t an

d 2

feet

dow

n,w

hat

will

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 d

esig

n of

a b

usin

ess

logo

cal

ls f

or lo

cati

ng t

he v

erti

ces

of a

tri

angl

e at

(1.5

,5),

(4,1

),an

d (1

,0)

on a

gri

d.If

des

ign

chan

ges

requ

ire

rota

ting

the

tri

angl

e 90

ºco

unte

rclo

ckw

ise,

wha

t w

ill t

he 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 (A

vera

ge)

Tran

sfor

mat

ions

with

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-4

4-4


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csTr

ansf

orm

atio

ns w

ith M

atric

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-Hi

ll19

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

ucti

on t

o L

esso

n 4-

4 at

the

top

of

page

175

in y

our

text

book

.

Des

crib

e ho

w y

ou c

an c

hang

e th

e or

ient

atio

n of

a f

igur

e w

itho

ut c

hang

ing

its

size

or

shap

e.

Flip

(or r

efle

ct) t

he fi

gure

ove

r a li

ne.

Rea

din

g t

he

Less

on

1.a.

Wri

te t

he v

erte

x m

atri

x fo

r th

e qu

adri

late

ral A

BC

Dsh

own

in

the

grap

h at

the

rig

ht.

b.W

rite

the

ver

tex

mat

rix

that

rep

rese

nts

the

posi

tion

of t

he q

uadr

ilate

ral A

'B'C

'D'

that

resu

lts

whe

n qu

adri

late

ral A

BC

Dis

tra

nsla

ted

3 un

its

to t

he r

ight

and

2 u

nits

dow

n.

2.D

escr

ibe

the

tran

sfor

mat

ion

that

cor

resp

onds

to

each

of

the

follo

win

g m

atri

ces.

a.b.

coun

terc

lock

wis

e ro

tatio

n tra

nsla

tion

4 un

its d

own

and

abou

t the

orig

in o

f 180

(3

units

to th

e rig

ht

c.d.

refle

ctio

n ov

er th

e y-

axis

refle

ctio

n ov

er 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

whi

ch o

f th

e re

flec

tion

mat

rice

s co

rres

pond

s to

ref

lect

ion

over

the

x-a

xis.

Sam

ple

answ

er:T

he o

nly

elem

ents

use

d in

the

refle

ctio

n m

atric

es a

re 0

,1,

and

#1.

For s

uch

a 2

!2

mat

rix M

to h

ave

the

prop

erty

that

M

&"

,the

ele

men

ts in

the

top

row

mus

t be

1 an

d 0

(in th

at

orde

r),an

d el

emen

ts in

the

botto

m ro

w m

ust b

e 0

and

#1

(in th

at o

rder

).

⎡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#

3⎦x

y

O

A

B

CD

©G

lenc

oe/M

cGra

w-Hi

ll19

2G

lenc

oe A

lgeb

ra 2

Prop

ertie

s 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

the

elem

ents

of

a ro

w (

or c

olum

n) a

re z

ero,

the

valu

e of

the

de

term

inan

t is

zer

o.

#0

(a"

0) !

(0 "

b) #

0

•M

ulti

plyi

ng a

ll th

e el

emen

ts o

f a

row

(or

col

umn)

by

a co

nsta

nt is

equ

ival

ent

to m

ulti

plyi

ng t

he v

alue

of

the

dete

rmin

ant

by t

he c

onst

ant.

3#

3[4(

3) !

5(!

1)]#

513[

12 $

5]#

5112

(3) !

5(!

3)#

51

•If

tw

o ro

ws

(or

colu

mns

) ha

ve e

qual

cor

resp

ondi

ng e

lem

ents

,the

val

ue o

f th

ede

term

inan

t is

zer

o.

#0

5(!

3) !

(!3)

(5) #

0

•T

he v

alue

of

a de

term

inan

t is

unc

hang

ed if

any

mul

tipl

e of

a r

ow (

or c

olum

n)

is a

dded

to

corr

espo

ndin

g el

emen

ts o

f an

othe

r ro

w (

or c

olum

n).

#4(

5) !

2(!

3)#

6(5)

!2(

2)#

2620

$6

#26

30 !

4#

26(R

ow 2

is a

dded

to ro

w 1.

)

•If

tw

o ro

ws

(or

colu

mns

) ar

e in

terc

hang

ed,t

he s

ign

of t

he d

eter

min

ant

is c

hang

ed.

#4(

8) !

(!3)

(5)#

![(!

3)(5

) !4(

8)]#

4732

$15

#47

![!

15 !

32]#

47

•T

he v

alue

of

the

dete

rmin

ant

is u

ncha

nged

if r

ow 1

is in

terc

hang

ed w

ith

colu

mn

1,an

d ro

w 2

is in

terc

hang

ed w

ith

colu

mn

2.T

he r

esul

t is

cal

led

the

tran

spos

e.

#5(

4) !

3(!

7)#

5(4)

!(!

7)(3

)#20

$21

#41

20 $

21#

41

Exe

rcis

es 1

–6

Ver

ify

each

pro

pert

y ab

ove

by e

valu

atin

g th

e gi

ven

dete

rmin

ants

and

giv

e an

othe

r ex

ampl

e of

the

pro

pert

y.Ex

ampl

es 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

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-4

4-4



Stu

dy

Gu

ide

and I

nte

rven

tion

Dete

rmin

ants

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-5

4-5

©G

lenc

oe/M

cGra

w-Hi

ll19

3G

lenc

oe A

lgeb

ra 2

Lesson 4-5

Det

erm

inan

ts o

f 2

!2

Mat

rice

s

Seco

nd-O

rder

Det

erm

inan

tFo

r the

mat

rix

, the

det

erm

inan

t 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

cd

ab

cd

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-Hi

ll19

4G

lenc

oe A

lgeb

ra 2

Det

erm

inan

ts o

f 3

!3

Mat

rice

s

Third

-Ord

er D

eter

min

ants

#a

"b

$c

Area

of a

Tria

ngle

The

area

of a

tria

ngle

hav

ing

verti

ces

(a, b

), (c

, d) a

nd (e

, f) i

s | A

| , wh

ere

A#

.

Eva

luat

e .

#4

"5

"2

Third

-ord

er d

eter

min

ant

#4(

18 "

0) "

5(6

"0)

"2(

"3

"6)

Eval

uate

2 !

2 de

term

inan

ts.

#4(

18)

"5(

6) "

2("

9)Si

mpl

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

iang

le w

ith

vert

ices

X(2

,"3)

,Y(7

,4),

and

Z("

5,5)

.44

.5 s

quar

e 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

gh

df

gi

ef

hi

ab

c

de

f

gh

i

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Dete

rmin

ants

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-5

4-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Dete

rmin

ants

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-5

4-5

©G

lenc

oe/M

cGra

w-Hi

ll19

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-Hi

ll19

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

ngle

who

se v

erti

ces

have

coo

rdin

ates

(3,

5),(

6,"

5),

and

("4,

10).

27.5

uni

ts2

26.L

AN

D M

AN

AG

EMEN

TA

fis

h an

d w

ildlif

e m

anag

emen

t or

gani

zati

on u

ses

a G

IS(g

eogr

aphi

c in

form

atio

n sy

stem

) to

sto

re a

nd a

naly

ze d

ata

for

the

parc

els

of la

nd it

man

ages

.All

of t

he p

arce

ls a

re m

appe

d on

a g

rid

in w

hich

1 u

nit

repr

esen

ts 1

acr

e.If

the

coor

dina

tes

of t

he c

orne

rs o

f a

parc

el a

re ("

8,10

),(6

,17)

,and

(2,

"4)

,how

man

yac

res

is t

he p

arce

l?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 (A

vera

ge)

Dete

rmin

ants

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-5

4-5



Rea

din

g t

o L

earn

Math

emati

csDe

term

inan

ts

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-5

4-5

©G

lenc

oe/M

cGra

w-Hi

ll19

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

ucti

on t

o L

esso

n 4-

5 at

the

top

of

page

182

in y

our

text

book

.

In t

his

less

on,y

ou w

ill le

arn

how

to

find

the

area

of a

tri

angl

e if

you

know

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 t

he a

rea

of t

he B

erm

uda

Tri

angl

e.Sa

mpl

e an

swer

:Us

e th

e m

ap.C

hoos

e an

y si

de o

f the

tria

ngle

as

the

base

,and

mea

sure

this

sid

e w

ith a

rule

r.M

ultip

ly th

is le

ngth

by

the

scal

efa

ctor

for t

he m

ap.N

ext,

draw

a s

egm

ent f

rom

the

oppo

site

vert

ex p

erpe

ndic

ular

to th

e ba

se.M

easu

re th

is s

egm

ent,

and

mul

tiply

its

leng

th b

y th

e sc

ale

for t

he m

ap.F

inal

ly,fin

d th

e ar

ea b

y us

ing

the

form

ula

A"

bh.

Rea

din

g t

he

Less

on

1.In

dica

te w

heth

er e

ach

of t

he f

ollo

win

g st

atem

ents

is t

rue

or f

alse

.

a.E

very

mat

rix

has

a de

term

inan

t.fa

lse

b.If

bot

h ro

ws

of a

2 !

2 m

atri

x ar

e id

enti

cal,

the

dete

rmin

ant

of t

he m

atri

x w

ill b

e 0.

true

c.E

very

ele

men

t of

a 3

!3

mat

rix

has

a m

inor

.tru

ed.

In o

rder

to

eval

uate

a t

hird

-ord

er d

eter

min

ant

by e

xpan

sion

by

min

ors

it is

nec

essa

ryto

fin

d th

e m

inor

of

ever

y el

emen

t of

the

mat

rix.

fals

ee.

If y

ou e

valu

ate

a th

ird-

orde

r de

term

inan

t by

exp

ansi

on a

bout

the

sec

ond

row

,the

posi

tion

sig

ns y

ou w

ill u

se a

re "

$"

.tru

e

2.Su

ppos

e th

at t

rian

gle

RS

Tha

s ve

rtic

es R

("2,

5),S

(4,1

),an

d T

(0,6

).

a.W

rite

a d

eter

min

ant

that

cou

ld b

e us

ed in

fin

ding

the

are

a of

tri

angl

e R

ST

.

b.E

xpla

in h

ow y

ou w

ould

use

the

det

erm

inan

t yo

u w

rote

in p

art

ato

fin

d th

e ar

ea o

fth

e tr

iang

le.

Sam

ple

answ

er:E

valu

ate

the

dete

rmin

ant a

nd m

ultip

ly th

e re

sult

by

.The

n ta

ke th

e ab

solu

te v

alue

to m

ake

sure

the

final

ans

wer

is

pos

itive

.

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 t

o br

eak

it d

own

into

ste

ps.W

rite

alis

t of

ste

ps f

or e

valu

atin

g a

thir

d-or

der

dete

rmin

ant

usin

g ex

pans

ion

by m

inor

s.Sa

mpl

e an

swer

:1.C

hoos

e a

row

of t

he m

atrix

.2.F

ind

the

posi

tion

sign

sfo

r the

row

you

hav

e ch

osen

.3.F

ind

the

min

or o

f eac

h el

emen

t in

that

row.

4.M

ultip

ly e

ach

elem

ent b

y its

pos

ition

sig

n an

d by

its

min

or.5

.Add

the

resu

lts.1 $ 2

⎢#2

51⎥

⎢4

11⎥

⎢0

61⎥

1 $ 2

©G

lenc

oe/M

cGra

w-Hi

ll19

8G

lenc

oe A

lgeb

ra 2

Mat

rix M

ultip

licat

ion

A f

urni

ture

man

ufac

ture

r m

akes

uph

olst

ered

cha

irs

and

woo

d ta

bles

.Mat

rix

Ash

ows

the

num

ber

of h

ours

spe

nt o

n ea

ch it

em b

y th

ree

diff

eren

t w

orke

rs.O

ne

day

the

fact

ory

rece

ives

an

orde

r fo

r 10

cha

irs

and

3 ta

bles

.Thi

s is

sho

wn

in

mat

rix

B.

#A

#B

[10

3]#

[10(

4) $

3(18

)10

(2) $

3(15

)10

(12)

$3(

0)] #

[94

6512

0]

The

pro

duct

of t

he t

wo

mat

rice

s sh

ows

the

num

ber

of h

ours

nee

ded

for

each

ty

pe o

f wor

ker

to c

ompl

ete

the

orde

r:94

hou

rs fo

r w

oodw

orki

ng,6

5 ho

urs

for

finis

hing

,and

120

hou

rs fo

r up

hols

teri

ng.

To f

ind

the

tota

l lab

or c

ost,

mul

tipl

y by

a m

atri

x th

at s

how

s th

e ho

urly

rat

e fo

r ea

ch w

orke

r:$1

5 fo

r w

oodw

orki

ng,$

9 fo

r fi

nish

ing,

and

$12

for

upho

lste

ring

.

C#

[94

6512

0] #

[94(

15)

$65

(9)

$12

0(12

)] #

$343

5

Use

mat

rix

mu

ltip

lica

tion

to

solv

e th

ese

pro

blem

s.A

can

dy c

ompa

ny p

acka

ges

cara

mel

s,ch

ocol

ates

,and

har

d ca

ndy

in t

hree

di

ffer

ent

asso

rtm

ents

:tra

diti

onal

,del

uxe,

and

supe

rb.F

or e

ach

type

of

cand

y th

e ta

ble

belo

w g

ives

the

num

ber

in e

ach

asso

rtm

ent,

the

num

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

The

com

pany

rec

eive

s an

ord

er f

or 3

00 t

radi

tion

al,1

80 d

elux

e an

d10

0 su

perb

ass

ortm

ents

.

1.F

ind

the

num

ber

of e

ach

type

of

cand

y ne

eded

to

fill

the

orde

r.73

80 c

aram

els;

7540

cho

cola

tes;

6680

har

d ca

ndie

s

2.F

ind

the

tota

l num

ber

of C

alor

ies

in e

ach

type

of

asso

rtm

ent.

1990

-trad

ition

al;2

400-

delu

xe;3

090-

supe

rb

3.F

ind

the

cost

of

prod

ucti

on f

or e

ach

type

of

asso

rtm

ent.

$3.0

4-tra

ditio

nal;

$3.5

2-de

luxe

;$4.

98-s

uper

b

4.F

ind

the

cost

to

fill

the

orde

r.$2

043.

60

15

9

12

42

12

18

150

chai

rta

ble

num

ber o

rder

ed[

103

]

hour

swo

odwo

rker

finsh

erup

holst

erer

chai

r 4

212

ta

ble

1815

0

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-5

4-5


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Cram

er’s

Rul

e

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-6

4-6

©G

lenc

oe/M

cGra

w-Hi

ll19

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

ving

sys

tem

sof

equ

atio

ns.

Cram

er’s

Rul

e fo

r

The

solu

tion

of th

e lin

ear s

yste

m o

f equ

atio

nsax

$by

#e

Two-

Varia

ble

Syst

ems

cx$

dy#

f

is (x

, y) w

here

x#

, y

#, a

nd

(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#

Cram

er’s

Rule

y#

#

a#

5, b

#"

10, c

#10

, d#

25, e

# 8

, f#

"2

#

#

Eval

uate

eac

h de

term

inan

t.#

#or

Si

mpl

ify.

#"

or "

The

sol

utio

n is

!,"

".

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

$18

y#

129x

$6y

#"

75x

"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

cf

$ ab

cd

eb

fd

$ ab

cd

ab

cd

ae

cf

$ ab

cd

eb

fd

$ ab

cd

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-Hi

ll20

0G

lenc

oe A

lgeb

ra 2

Syst

ems

of

Thre

e Li

nea

r Eq

uat

ion

s

The

solu

tion

of th

e sy

stem

who

se e

quat

ions

are

ax$

by$

cz#

j

Cram

er’s

Rul

e fo

r

dx$

ey$

fz#

k

Thre

e-Va

riabl

e Sy

stem

s

gx$

hy$

iz#

l

is (x

, y, z

) whe

re x

#, y

#, a

nd 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

the

coe

ffic

ient

s an

d co

nsta

nts

from

the

equ

atio

ns t

o fo

rm t

he d

eter

min

ants

.The

nev

alua

te e

ach

dete

rmin

ant.

x#

y

#z

#

#or

#

or "

#or

The

sol

utio

n is

!,"

,".

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

gh

l $ a

bc

de

f g

hi

aj

c d

kf

gl

i $ a

bc

de

f g

hi

jb

c k

ef

lh

i $ a

bc

de

f g

hi

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Cram

er’s

Rul

e

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-6

4-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Cram

er’s

Rul

e

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-6

4-6

©G

lenc

oe/M

cGra

w-Hi

ll20

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

The

tw

o si

des

of a

n an

gle

are

cont

aine

d in

the

line

s w

hose

equ

atio

ns a

re3x

$2y

#4

and

x"

3y#

5.F

ind

the

coor

dina

tes

of t

he v

erte

x of

the

ang

le.

(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

#4

(1,#

1,#

1)

©G

lenc

oe/M

cGra

w-Hi

ll20

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

The

tw

o si

des

of a

n an

gle

are

cont

aine

d in

the

line

s w

hose

equ

atio

ns a

re

x"

y#

6 an

d 2x

$y

#1.

Fin

d th

e co

ordi

nate

s of

the

ver

tex

of t

he a

ngle

.(2

,#3)

17.G

EOM

ETRY

Tw

o si

des

of a

par

alle

logr

am a

re c

onta

ined

in t

he li

nes

who

se e

quat

ions

are

0.2x

"0.

5y#

1 an

d 0.

02x

"0.

3y#

"0.

9.F

ind

the

coor

dina

tes

of a

ver

tex

of t

hepa

ralle

logr

am.

(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#

"3

23.3

x"

2y$

5z#

34c

$d

$5e

#"

93j

$2k

$4m

#5

2x$

2y"

4z#

3c

$2d

"e

#12

"4j

"k

$2m

#4

"5x

$10

y$

7z#

"3

(2,3

,#4)

(#1,

2,1)

(1.2

,0.3

,0)

24.L

AN

DSC

API

NG

A m

emor

ial g

arde

n be

ing

plan

ted

in

fron

t of

a m

unic

ipal

libr

ary

will

con

tain

thr

ee c

ircu

lar

beds

tha

t ar

e ta

ngen

t to

eac

h ot

her.

A la

ndsc

ape

arch

itec

t ha

s pr

epar

ed a

ske

tch

of t

he d

esig

n fo

r th

e ga

rden

usi

ng

CA

D (

com

pute

r-ai

ded

draf

ting

) so

ftw

are,

as s

how

n at

the

righ

t.T

he c

ente

rs o

f th

e th

ree

circ

ular

bed

s ar

e re

pres

ente

dby

poi

nts

A,B

,and

C.T

he d

ista

nce

from

Ato

Bis

15

feet

,th

e di

stan

ce f

rom

Bto

Cis

13

feet

,and

the

dis

tanc

e fr

om

Ato

Cis

16

feet

.Wha

t is

the

rad

ius

of e

ach

of t

he c

ircu

lar

beds

?ci

rcle

A:9

ft,c

ircle

B:6

ft,c

ircle

C:7

ft

A

BC

13 ft

16 ft

15 ft

4 % 3

4 $ 31 $ 3

Pra

ctic

e (A

vera

ge)

Cram

er’s

Rul

e

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-6

4-6


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csCr

amer

’s R

ule

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-6

4-6

©G

lenc

oe/M

cGra

w-Hi

ll20

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

equ

atio

ns?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 4-

6 at

the

top

of

page

189

in y

our

text

book

.

A t

rian

gle

is b

ound

ed b

y th

e x-

axis

,the

line

y#

x,an

d th

e lin

e

y#

"2x

$10

.Wri

te t

hree

sys

tem

s of

equ

atio

ns t

hat

you

coul

d us

e to

fin

dth

e th

ree

vert

ices

of

the

tria

ngle

.(D

o no

t ac

tual

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.Su

ppos

e th

at y

ou a

re a

sked

to

solv

e th

e fo

llow

ing

syst

em o

f eq

uati

ons

by C

ram

er’s

Rul

e.

3x$

2y#

72x

"3y

#22

Wit

hout

act

ually

eva

luat

ing

any

dete

rmin

ants

,ind

icat

e w

hich

of

the

follo

win

g ra

tios

of

dete

rmin

ants

giv

es t

he c

orre

ct v

alue

for

x.

B

A.

B.

C.

2.In

you

r te

xtbo

ok,t

he s

tate

men

ts o

f C

ram

er’s

Rul

e fo

r tw

o va

riab

les

and

thre

e va

riab

les

spec

ify

that

the

det

erm

inan

t fo

rmed

fro

m t

he c

oeff

icie

nts

of t

he v

aria

bles

can

not

be 0

.If

the

det

erm

inan

t is

zer

o,w

hat

do y

ou k

now

abo

ut t

he s

yste

m a

nd it

s so

luti

ons?

The

syst

em c

ould

be

a de

pend

ent s

yste

m a

nd h

ave

infin

itely

man

yso

lutio

ns,o

r it c

ould

be

an in

cons

iste

nt s

yste

m a

nd h

ave

no s

olut

ions

.

Hel

pin

g Y

ou

Rem

emb

er

3.So

me

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

sys

tem

of

two

linea

r eq

uati

ons

by C

ram

er’s

Rul

e.W

hat

is a

goo

d w

ay t

ore

mem

ber

this

?

Sam

ple

answ

er:L

et D

be th

e de

term

inan

t of t

he c

oeffi

cien

ts.L

et D

xbe

the

dete

rmin

ant f

orm

ed b

y re

plac

ing

the

first

col

umn

of D

with

the

cons

tant

sfro

m th

e rig

ht-h

and

side

of t

he s

yste

m,a

nd le

t Dy

be th

e de

term

inan

tfo

rmed

by

repl

acin

g th

e se

cond

col

umn

of D

with

the

cons

tant

s.Th

en

x"

and

y"

.D y$ D

D x$ 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-Hi

ll20

4G

lenc

oe A

lgeb

ra 2

Com

mun

icat

ions

Net

wor

ksT

he d

iagr

am a

t th

e ri

ght

repr

esen

ts a

com

mun

icat

ions

net

wor

k lin

king

fiv

e co

mpu

ter

rem

ote

stat

ions

.The

arr

ows

indi

cate

the

dire

ctio

n in

whi

ch s

igna

ls c

an b

e tr

ansm

itte

d an

d re

ceiv

ed b

y ea

chco

mpu

ter.

We

can

gene

rate

a m

atri

x to

des

crib

e th

is n

etw

ork.

A#

Mat

rix

Ais

a c

omm

unic

atio

ns n

etw

ork

for

dire

ct c

omm

unic

atio

n.Su

ppos

e yo

u w

ant

to s

end

a m

essa

ge f

rom

one

com

pute

r to

ano

ther

usi

ng e

xact

ly o

ne o

ther

co

mpu

ter

as a

rel

ay p

oint

.It

can

be s

how

n th

at t

he e

ntri

es o

f m

atri

x A

2re

pres

ent

the

num

ber

of w

ays

to s

end

a m

essa

ge f

rom

one

poi

nt t

o an

othe

r by

goi

ng t

hrou

gh

a th

ird

stat

ion.

For

exam

ple,

a m

essa

ge m

ay b

e se

nt f

rom

sta

tion

1 t

o st

atio

n 5

by

goin

g th

roug

h st

atio

n 2

or s

tati

on 4

on

the

way

.The

refo

re,A

2 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 w

ays

the

mes

sage

s ca

n b

e se

nt

for

each

mat

rix.

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

0⎤⎢0

10

00⎥

⎢11

10

0⎥⎢1

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

0⎤⎢1

00

00⎥

⎢10

01

0⎥⎢1

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⎦

25

1 34

2 61

3

45

1

23

4

Aga

in,c

ompa

re t

he e

ntri

es o

f m

atri

x A

2to

the

com

mun

icat

ions

dia

gram

to

veri

fy t

hat

the

entr

ies

are

corr

ect.

Mat

rix

A2

repr

esen

ts u

sing

exa

ctly

one

rel

ay.

to c

ompu

ter j

11

11

2fro

m1

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 fr

om c

ompu

ter

ito

com

pute

r jd

irec

tly.

Com

pare

the

entr

ies

of m

atri

x A

to t

he d

iagr

am t

o ve

rify

the

ent

ries

.For

exam

ple,

ther

e is

one

way

to

send

a m

essa

ge f

rom

com

pute

r 3

to c

ompu

ter

4,so

A3,

4#

1.A

com

pute

r ca

nnot

sen

d a

mes

sage

to it

self,

so A

1,1

#0.

to c

ompu

ter j

01

01

0fro

m0

00

11

com

pute

r i1

00

10

11

10

1 0

10

10

12

34

5

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-6

4-6



Stu

dy

Gu

ide

and I

nte

rven

tion

Iden

tity

and

Inve

rse

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-7

4-7

©G

lenc

oe/M

cGra

w-Hi

ll20

5G

lenc

oe A

lgeb

ra 2

Lesson 4-7

Iden

tity

an

d In

vers

e M

atri

ces

The

iden

tity

mat

rix

for

mat

rix

mul

tipl

icat

ion

is a

squa

re m

atri

x w

ith

1s f

or e

very

ele

men

t of

the

mai

n di

agon

al a

nd z

eros

els

ewhe

re.

Iden

tity

Mat

rixIf

Ais

an n

!n

mat

rix a

nd I

is th

e id

entit

y m

atrix

,fo

r Mul

tiplic

atio

nth

en A

&I#

Aan

d I&

A#

A.

If a

n n

!n

mat

rix

Aha

s an

inve

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

Sinc

e 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

noye

s

7.an

d 8.

and

9.an

d

noye

sno

10.

and

11.

and

12.

and

noye

sye

s

"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

6⎦Ex

ampl

eEx

ampl

e

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-Hi

ll20

6G

lenc

oe A

lgeb

ra 2

Fin

d In

vers

e M

atri

ces

Inve

rse

of a

2 !

2 M

atrix

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 in

vers

e.

Fin

d t

he

inve

rse

of N

".

Fir

st f

ind

the

valu

e of

the

det

erm

inan

t.

#7

"4

#3

Sinc

e th

e de

term

inan

t do

es n

ot e

qual

0,N

"1

exis

ts.

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 in

vers

e ex

ists

4.5.

6.

#$1 2$

no in

vers

e ex

ists

⎡4

#2⎤

⎣#10

#8⎦

1 $ 12⎡

8#

5⎤⎣#

106⎦

8

2 1

04

36

48

6

5 1

08

⎡30

10⎤

⎣20

40⎦

1$ 10

00⎡1

#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

cd

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Iden

tity

and

Inve

rse

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-7

4-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Iden

tity

and

Inve

rse

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-7

4-7

©G

lenc

oe/M

cGra

w-Hi

ll20

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

#no

4.A

#,B

#ye

s

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 in

vers

e ex

ists

12.

13.

14.

no in

vers

e ex

ists

15.

16.

#

17.

#18

.no

inve

rse

exis

ts

19.

#20

.⎡2

0⎤⎣0

2⎦1 $ 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

1⎦1 $ 6

1"

1 3

3

⎡0

4⎤⎣#

6#

2⎦1 $ 24

"2

"4

6

09

3 6

2

⎡2

#1⎤

⎣#3

1⎦1

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-Hi

ll20

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

#no

2.X

#,Y

#ye

s

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 sq

uare

mat

rice

s ha

ve m

ulti

plic

ativ

e in

vers

es.

fals

e6.

All

squa

re m

atri

ces

have

mul

tipl

icat

ive

iden

titi

es.

true

Fin

d t

he

inve

rse

of e

ach

mat

rix,

if i

t ex

ists

.

7.8.

9.#

10.

11.

12.

no in

vers

e ex

ists

GEO

MET

RYF

or E

xerc

ises

13–

16,u

se t

he

figu

re a

t th

e ri

ght.

13.W

rite

the

ver

tex

mat

rix

Afo

r th

e re

ctan

gle.

14.U

se m

atri

x m

ulti

plic

atio

n to

fin

d B

Aif

B#

.

15.G

raph

the

ver

tice

s of

the

tra

nsfo

rmed

tri

angl

e on

the

pre

viou

s gr

aph.

Des

crib

e th

e tr

ansf

orm

atio

n.di

latio

n by

a s

cale

fact

or o

f 1.5

16.M

ake

a co

njec

ture

abo

ut w

hat

tran

sfor

mat

ion

B"

1de

scri

bes

on a

coo

rdin

ate

plan

e.

dila

tion

by a

sca

le fa

ctor

of

17.C

OD

ESU

se t

he a

lpha

bet

tabl

e be

low

and

the

inve

rse

of c

odin

g m

atri

x C

#

tode

code

thi

s m

essa

ge:

19|14

|11|13

|11|22

|55|65

|57|60

|2|1

|52|47

|33|51

|56|55

.

CHEC

K_YO

UR_A

NSW

ERS

CODE

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#

5⎤⎣1

2⎦1 $ 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 (A

vera

ge)

Iden

tity

and

Inve

rse

Mat

rices

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-7

4-7



Rea

din

g t

o L

earn

Math

emati

csId

entit

y an

d In

vers

e M

atric

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-7

4-7

©G

lenc

oe/M

cGra

w-Hi

ll20

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

ucti

on t

o L

esso

n 4-

7 at

the

top

of

page

195

in y

our

text

book

.

Ref

er t

o th

e co

de t

able

giv

en in

the

intr

oduc

tion

to

this

less

on.S

uppo

se t

hat

you

rece

ive

a m

essa

ge c

oded

by

this

sys

tem

as

follo

ws:

1612

51

195

25

1325

618

95

144.

Dec

ode

the

mes

sage

.Pl

ease

be

my

frien

d.

Rea

din

g t

he

Less

on

1.In

dica

te w

heth

er e

ach

of t

he f

ollo

win

g st

atem

ents

is t

rue

or f

alse

.

a.E

very

ele

men

t of

an

iden

tity

mat

rix

is 1

.fa

lse

b.T

here

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 t

heir

pro

duct

is t

he id

enti

ty m

atri

x.tru

ed.

If M

is a

mat

rix,

M"

1re

pres

ents

the

rec

ipro

cal o

f M.

fals

ee.

No

3 !

2 m

atri

x ha

s an

inve

rse.

true

f.E

very

squ

are

mat

rix

has

an in

vers

e.fa

lse

g.If

the

tw

o co

lum

ns o

f a

2 !

2 m

atri

x ar

e id

enti

cal,

the

mat

rix

does

not

hav

e an

inve

rse.

true

2.E

xpla

in h

ow t

o fi

nd t

he in

vers

e of

a 2

!2

mat

rix.

Do

not

use

any

spec

ial m

athe

mat

ical

sym

bols

in y

our

expl

anat

ion.

Sam

ple

answ

er:F

irst f

ind

the

dete

rmin

ant o

f the

mat

rix.I

f it i

s ze

ro,t

hen

the

mat

rix h

as n

o in

vers

e.If

the

dete

rmin

ant i

s no

t zer

o,fo

rm a

new

mat

rix a

s fo

llow

s.In

terc

hang

e th

e to

p le

ft an

d bo

ttom

righ

t ele

men

ts.

Chan

ge th

e si

gns

but n

ot th

e po

sitio

ns o

f the

oth

er tw

o el

emen

ts.

Mul

tiply

the

resu

lting

mat

rix b

y th

e re

cipr

ocal

of t

he d

eter

min

ant o

f the

orig

inal

mat

rix.T

he re

sulti

ng m

atrix

is th

e in

vers

e of

the

orig

inal

mat

rix.

Hel

pin

g Y

ou

Rem

emb

er

3.O

ne w

ay t

o re

mem

ber

som

ethi

ng is

to

expl

ain

it t

o an

othe

r pe

rson

.Sup

pose

tha

t yo

u ar

est

udyi

ng w

ith

a cl

assm

ate

who

is h

avin

g tr

oubl

e re

mem

beri

ng h

ow t

o fi

nd t

he in

vers

e of

a 2

!2

mat

rix.

He

rem

embe

rs h

ow t

o m

ove

elem

ents

and

cha

nge

sign

s in

the

mat

rix,

but

thin

ks t

hat

he s

houl

d m

ulti

ply

by t

he d

eter

min

ant

of t

he o

rigi

nal m

atri

x.H

ow c

anyo

u he

lp h

im r

emem

ber

that

he

mus

t m

ulti

ply

by t

he r

ecip

roca

lof

thi

s de

term

inan

t?

Sam

ple

answ

er:I

f the

det

erm

inan

t of t

he m

atrix

is 0

,its

reci

proc

al is

unde

fined

.Thi

s ag

rees

with

the

fact

that

if th

e de

term

inan

t of a

mat

rix is

0,th

e m

atrix

doe

s no

t hav

e an

inve

rse.

©G

lenc

oe/M

cGra

w-Hi

ll21

0G

lenc

oe A

lgeb

ra 2

Perm

utat

ion

Mat

rices

A p

erm

utat

ion

mat

rix

is a

squ

are

mat

rix

in w

hich

ea

ch r

ow a

nd e

ach

colu

mn

has

one

entr

y th

at is

1.

All

the

othe

r en

trie

s ar

e 0.

Fin

d th

e in

vers

e of

ape

rmut

atio

n m

atri

x in

terc

hang

ing

the

row

s an

dco

lum

ns.F

or e

xam

ple,

row

1 is

inte

rcha

nged

wit

hco

lum

n 1,

row

2 is

inte

rcha

nged

wit

h co

lum

n 2.

Pis

a 4

!4

perm

utat

ion

mat

rix.

P"

1is

the

inve

rse

of P

.

Sol

ve e

ach

pro

blem

.

1.T

here

is ju

st o

ne 2

!2

perm

utat

ion

2.F

ind

the

inve

rse

of t

he m

atri

x yo

u w

rote

m

atri

x th

at is

not

als

o an

iden

tity

in

Exe

rcis

e 1.

Wha

t do

you

not

ice?

mat

rix.

Wri

te t

his

mat

rix.

The

two

mat

rices

are

the

sam

e.

3.Sh

ow t

hat

the

two

mat

rice

s in

Exe

rcis

es 1

and

2 a

re in

vers

es.

#

4.W

rite

the

inve

rse

of t

his

mat

rix.

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.Pe

rmut

atio

n m

atri

ces

can

be u

sed

to w

rite

and

dec

iphe

r co

des.

To s

ee h

ow

this

is d

one,

use

the

mes

sage

mat

rix

Man

d m

atri

x B

from

pro

blem

4.F

ind

mat

rix

Cso

tha

t C

equa

ls t

he p

rodu

ct M

B.U

se t

he r

ules

bel

ow.

0 tim

es a

lette

r #0

1 tim

es a

lette

r #th

e sa

me

lette

r0

plus

a le

tter #

the

sam

e le

tter

7.N

ow f

ind

the

prod

uct

CB

"1 .

Wha

t do

you

not

ice?

#

#

Mul

tiply

ing

Mby

Ben

code

s th

e m

essa

ge.T

o de

ciph

er,m

ultip

ly b

y B

!1 .

⎡SH

E⎤

⎢SA

W⎥

⎣HI

M⎦

⎡01

0⎤⎢0

01⎥

⎣10

0⎦

⎡HE

S⎤

⎢AW

S⎥

⎣I

MH

⎦

⎡01

0⎤⎢0

01⎥

⎣10

0⎦

⎡0 &

0 %

0 &

0 %

1 &

10

&1

%0

&0

%1

&0

0 &

0 %

0 &

1 %

1 &

0⎤⎢1

&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

0⎤⎢0

01⎥

⎣10

0⎦0

01

10

00

10

⎡10⎤

⎣01⎦

⎡0 &

1 %

1 &

10

&1

%1

&0⎤

⎣1 &

0 %

0 &

11

&1

%0

&0⎦

⎡01⎤

⎣10⎦

⎡01⎤

⎣10⎦

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

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

⎦S

HE

SA

W

HI

M


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Usin

g M

atric

es to

Sol

ve S

yste

ms

of E

quat

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-8

4-8

©G

lenc

oe/M

cGra

w-Hi

ll21

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

uati

ons

cons

ists

of

the

prod

uct

of t

he c

oeff

icie

nt a

nd v

aria

ble

mat

rice

s on

the

left

and

the

con

stan

t m

atri

x on

the

righ

t of

the

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

,and

cons

tant

mat

rice

s.

&#

12

"8

x

y

3"

7 1

5

b.2x

"y

$3z

#"

7x

$3y

"4z

#15

7x$

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

"y

$7z

#32

5x"

y$

3z#

15x

$3y

"2z

#"

18x

$2y

"6z

#25

2x$

4y"

3z#

12

& "

&

"

9.4x

"3y

"z

#"

100

10.x

"3y

$7z

#27

2x$

y"

3z#

"64

2x$

y"

5z#

485x

$3y

"2z

#8

4x"

2y$

3z#

72

& "

&

"

11.2

x$

3y"

9z#

"10

812

.z#

45 "

3x$

2yx

$5z

#40

$2y

2x$

3y"

z#

603x

$5y

#89

$4z

x#

4y"

2z$

120

& "

&

" ⎡

45⎤

⎢60

⎥⎣1

20⎦

⎡x⎤

⎢y⎥

⎣z⎦

⎡3#

21⎤

⎢23

#1⎥

⎣1#

42⎦

⎡#10

8⎤⎢

40⎥

⎣89

⎦

⎡x⎤

⎢y⎥

⎣z⎦

⎡23

#9⎤

⎢1#

25⎥

⎣35

#4⎦

⎡27⎤

⎢48⎥

⎣72⎦

⎡x⎤

⎢y⎥

⎣z⎦

⎡1#

37⎤

⎢21

#5⎥

⎣4#

23⎦

⎡#10

0⎤⎢

#64

⎥⎣

8⎦

⎡x⎤

⎢y⎥

⎣z⎦

⎡4#

3#

1⎤⎢2

1#

3⎥⎣5

3#

2⎦

⎡32

⎤⎢#

18⎥

⎣12

⎦

⎡x⎤

⎢y⎥

⎣z⎦

⎡5#

17⎤

⎢13

#2⎥

⎣24

#3⎦

⎡12⎤

⎢15⎥

⎣25⎦

⎡x⎤

⎢y⎥

⎣z⎦

⎡21

7⎤⎢5

#1

3⎥⎣1

2#

6⎦

⎡24⎤

⎣80⎦

⎡x⎤

⎣y⎦

⎡3#

1⎤⎣2

3⎦⎡1

8⎤⎣2

5⎦⎡x

⎤⎣y

⎦⎡5

2⎤⎣1

4⎦⎡2

0⎤⎣#

8⎦⎡x

⎤⎣y

⎦⎡4

#6⎤

⎣31⎦

⎡15

⎤⎣#

10⎦

⎡x⎤

⎣y⎦

⎡7#

2⎤⎣3

1⎦⎡1

8⎤⎣1

2⎦⎡x

⎤⎣y

⎦⎡4

#3⎤

⎣12⎦

⎡8⎤

⎣#12

⎦⎡x

⎤⎣y

⎦⎡2

1⎤⎣5

#3⎦

©G

lenc

oe/M

cGra

w-Hi

ll21

2G

lenc

oe A

lgeb

ra 2

Solv

e Sy

stem

s o

f Eq

uat

ion

sU

se in

vers

e m

atri

ces

to s

olve

sys

tem

s of

equ

atio

nsw

ritt

en a

s m

atri

x eq

uati

ons.

Solv

ing

Mat

rix E

quat

ions

If AX

#B,

then

X#

A"1 B

, whe

re A

is th

e co

effic

ient

mat

rix,

Xis

the

varia

ble

mat

rix, a

nd B

is th

e co

nsta

nt m

atrix

.

Sol

ve

".

In t

he m

atri

x eq

uati

on A

#,X

#,a

nd B

#.

Step

1F

ind

the

inve

rse

of t

he c

oeff

icie

nt m

atri

x.

A"

1#

or

.

Step

2M

ulti

ply

each

sid

e of

the

mat

rix

equa

tion

by

the

inve

rse

mat

rix.

& &

#&

Mul

tiply

each

sid

e by

A"

1 .

& #

Mul

tiply

mat

rices

.

#Si

mpl

ify.

The

sol

utio

n is

(2,

"2)

.

Sol

ve e

ach

mat

rix

equ

atio

n o

r sy

stem

of

equ

atio

ns

by u

sin

g in

vers

e m

atri

ces.

1.&

#2.

& #

3.&

#

(5,#

3)no

sol

utio

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

"4

"8

6

12

"2

18

x

y

24

3"

1

2

"2

x

y

16

"

16

1 % 8x

y

10

01

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

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Usin

g M

atric

es to

Sol

ve S

yste

ms

of E

quat

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-8

4-8

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Usin

g M

atric

es to

Sol

ve S

yste

ms

of E

quat

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-8

4-8

©G

lenc

oe/M

cGra

w-Hi

ll21

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 u

sin

g in

vers

e m

atri

ces.

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

#1⎥

⎣1

11⎦

⎡5⎤

⎣10⎦

⎡x⎤

⎣y⎦

⎡11⎤

⎣32⎦

⎡1⎤

⎣12⎦

⎡r⎤

⎣s⎦

⎡1#

1⎤⎣2

3⎦

⎡6⎤

⎣7⎦

⎡c⎤

⎣d⎦

⎡23⎤

⎣3#

4⎦⎡#

3⎤⎣

6⎦⎡m

⎤⎣n

⎦⎡1

3⎤⎣4

3⎦

⎡16⎤

⎣3⎦

⎡a⎤

⎣b⎦

⎡38⎤

⎣43⎦

⎡5⎤

⎣1⎦

⎡x⎤

⎣y⎦

⎡11⎤

⎣2#

1⎦

©G

lenc

oe/M

cGra

w-Hi

ll21

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 u

sin

g in

vers

e m

atri

ces.

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

line

flew

89

pass

enge

rs o

n a

com

mut

er f

light

fro

m B

osto

n to

New

Yor

k.So

me

of t

he p

asse

nger

s pa

id $

120

for

thei

rti

cket

s an

d th

e re

st p

aid

$230

for

the

ir t

icke

ts.T

he t

otal

cos

t of

all

of t

he t

icke

ts w

as$1

4,20

0.H

ow m

any

pass

enge

rs b

ough

t $1

20 t

icke

ts?

How

man

y bo

ught

$23

0 ti

cket

s?57

;32

18.N

UTR

ITIO

NA

sin

gle

dose

of

a di

etar

y su

pple

men

t co

ntai

ns 0

.2 g

ram

of

calc

ium

and

0.

2 gr

am o

f vi

tam

in C

.A s

ingl

e do

se o

f a

seco

nd d

ieta

ry s

uppl

emen

t co

ntai

ns 0

.1 g

ram

of

cal

cium

and

0.4

gra

m o

f vi

tam

in C

.If

a pe

rson

wan

ts t

o ta

ke 0

.6 g

ram

of

calc

ium

and

1.2

gram

s of

vit

amin

C,h

ow m

any

dose

s of

eac

h su

pple

men

t sh

ould

she

tak

e?2

dose

s of

the

first

sup

plem

ent a

nd 2

dos

es o

f the

sec

ond

supp

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⎤⎣#

2⎦⎡a

⎤⎣b

⎦⎡2

1⎤⎣3

2⎦

⎡#2⎤

⎣10⎦

⎡x⎤

⎣y⎦

⎡6#

2⎤⎣3

3⎦⎡

9⎤⎣#

13⎦

⎡x⎤

⎣y⎦

⎡#3

2⎤⎣

5#

3⎦Pra

ctic

e (A

vera

ge)

Usin

g M

atric

es to

Sol

ve S

yste

ms

of E

quat

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-8

4-8


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csUs

ing

Mat

rices

to S

olve

Sys

tem

s of

Equ

atio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-8

4-8

©G

lenc

oe/M

cGra

w-Hi

ll21

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

ucti

on t

o L

esso

n 4-

8 at

the

top

of

page

202

in y

our

text

book

.

Wri

te a

2 !

2 m

atri

x th

at s

umm

ariz

es t

he in

form

atio

n gi

ven

in t

hein

trod

ucti

on a

bout

the

foo

d an

d te

rrit

ory

requ

irem

ents

for

the

tw

o sp

ecie

s.

Rea

din

g t

he

Less

on

1.a.

Wri

te a

mat

rix

equa

tion

for

the

fol

low

ing

syst

em o

f eq

uati

ons.

3x$

5y#

10&

"

2x"

4y#

"7

b.E

xpla

in h

ow t

o us

e th

e m

atri

x eq

uati

on y

ou w

rote

abo

ve t

o so

lve

the

syst

em.U

se a

sfe

w m

athe

mat

ical

sym

bols

in y

our

expl

anat

ion

as y

ou c

an.D

o no

t ac

tual

ly s

olve

the

syst

em.

Sam

ple

answ

er:F

ind

the

inve

rse

of th

e 2

!2

mat

rix o

f coe

ffici

ents

.M

ultip

ly th

is in

vers

e by

the

2 !

1 m

atrix

of c

onst

ants

,with

the

2 !

2m

atrix

on

the

left.

The

prod

uct w

ill b

e a

2 !

1 m

atrix

.The

num

ber i

nth

e fir

st ro

w w

ill b

e th

e va

lue

of x

,and

the

num

ber i

n th

e se

cond

row

will

be

the

valu

e of

y.

2.W

rite

a s

yste

m o

f eq

uati

ons

that

cor

resp

onds

to

the

follo

win

g m

atri

x eq

uati

on.

& #

3x%

2y#

4z"

#2

2x#

y"

65y

%6z

"#

4

Hel

pin

g Y

ou

Rem

emb

er

3.So

me

stud

ents

hav

e tr

oubl

e re

mem

beri

ng h

ow t

o se

t up

a m

atri

x eq

uati

on t

o so

lve

asy

stem

of

linea

r eq

uati

ons.

Wha

t is

an

easy

way

to

rem

embe

r th

e or

der

in w

hich

to

wri

teth

e th

ree

mat

rice

s th

at m

ake

up t

he e

quat

ion?

Sam

ple

answ

er:J

ust r

emem

ber “

CVC”

for “

coef

ficie

nts,

varia

bles

,co

nsta

nts.”

The

varia

ble

mat

rix is

on

the

left

side

of t

he e

qual

s si

gn,j

ust

as th

e va

riabl

es a

re in

the

syst

em o

f lin

ear e

quat

ions

.The

con

stan

tm

atrix

is o

n th

e rig

ht s

ide

of th

e eq

uals

sig

n,ju

st a

s th

e co

nsta

nts

are

inth

e sy

stem

of l

inea

r 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-Hi

ll21

6G

lenc

oe A

lgeb

ra 2

Prop

ertie

s of

Mat

rices

Com

puti

ng w

ith

mat

rice

s is

dif

fere

nt f

rom

com

puti

ng w

ith

real

num

bers

.Sta

ted

belo

w a

re s

ome

prop

erti

es o

f th

e re

al n

umbe

r sy

stem

.Are

the

se a

lso

true

for

m

atri

ces?

In

the

prob

lem

s on

thi

s pa

ge,y

ou w

ill in

vest

igat

e th

is q

uest

ion.

For

all r

eal n

umbe

rs a

and

b,ab

#0

if a

nd o

nly

if a

#0

or b

#0.

Mul

tipl

icat

ion

is c

omm

utat

ive.

For

all r

eal n

umbe

rs a

and

b,ab

#ba

.

Mul

tipl

icat

ion

is a

ssoc

iati

ve.F

or a

ll re

al n

umbe

rs a

,b,a

nd c

,a(b

c) #

(ab)

c.

Use

th

e m

atri

ces

A,B

,an

d C

for

the

pro

blem

s.W

rite

wh

eth

er e

ach

st

atem

ent

is t

rue.

Ass

um

e th

at a

2-b

y-2

mat

rix

is t

he

0 m

atri

x if

an

d

only

if

all

of i

ts e

lem

ents

are

zer

o.

A#

B

#

C#

1.A

B#

0no

2.A

C#

0no

3.B

C#

0ye

s

AB#

AB

#

AB#

4.A

B#

BA

no5.

AC

#C

Ano

6.B

C#

CB

no

BA#

CA

#

AB#

So,A

B %

BA.

So,A

C %

CA.

So,B

C %

CB.

7.A

(BC

) #(A

B)C

yes

8.B

(CA

) #(B

C)A

yes

9.B

(AC

) #(B

A)C

yes

Both

pro

duct

s eq

ual

Both

pro

duct

s eq

ual

Both

pro

duct

s eq

ual

..

.

10.W

rite

a s

tate

men

t su

mm

ariz

ing

your

fin

ding

s ab

out

the

prop

erti

es o

f m

atri

xm

ulti

plic

atio

n.Ba

sed

on th

ese

exam

ples

,mat

rix m

ultip

licat

ion

is a

ssoc

iativ

e,bu

t not

com

mut

ativ

e.Tw

o m

atric

es m

ay h

ave

a pr

oduc

t of z

ero

even

if n

eith

er o

fth

e fa

ctor

s eq

uals

zer

o.

⎡#8

#16

⎤⎣

816

⎦⎡0

0⎤⎣0

0⎦⎡0

0⎤⎣0

0⎦

⎡#3

9⎤⎣#

13⎦

⎡15

21⎤

⎣5

7⎦⎡0

#8⎤

⎣08⎦

⎡00⎤

⎣00⎦

⎡10

20⎤

⎣6

12⎦

⎡2

#6⎤

⎣#2

6⎦

36

12

1

"3

"1

33

1 1

3

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

4-8

4-8

chapter 4 resource masters - math class · 2019-11-24 · ©glencoe/mcgraw-hill iv glencoe algebra...

Documents