section 4.1 matrix operations - edl · section 4.1 matrix operations objective(s): organize data...

Name: _______________________ Period: ______________________

Reflection: 1

Section 4.1 Matrix Operations Objective(s): Organize data using matrices. Simplify matrix expressions using the properties of matrices.

Essential Question: Is matrix addition commutative? Explain your answer.

Homework: Assignment 4.1. #1 – 21 in the homework packet.

Notes:

Vocabulary

A matrix is a rectangular arrangement of numbers (called entries) in rows (horizontal) and columns

(vertical); plural of matrix is matrices

The dimensions (size or order) of a matrix is the Number of Rows (by) Number of Columns.

A square matrix is a matrix with the same number of rows and columns.

Two matrices are equal if their dimensions are the same, and all of the corresponding entries are equal.

Organizing Data in a Matrix

Ex: Store A sells 550 DVDs, 420 video games, and 910 CDs on average every week. Store B sells 405

DVDs, 300 video games, and 1100 CDs on average every week. Use a matrix to organize this information.

State the dimensions of this matrix.

Solution:

DVDs VGs CDs

Store A 550 420 910

Store B 405 300 1100

The dimensions (order) of the matrix: 2 3

Determine the order (size) of the matrix.

Example 1: 1 2

4 7 The matrix is a ______ X ______.

Reflection: 2

Example 2:

11 0 3

2

24 12 7

3

75 0 0

2

The matrix is a ______ X ______.

Adding & Subtracting Matrices:

In order to add or subtract matrices, the dimensions must be the same.

Ex: Add the matrices: 4 3 1 8 1 5 .

Step One: Determine if the matrices can be added by checking the dimensions.

The matrices both have dimensions 1 3 , so they can be added.

Step Two: Add the corresponding entries. The sum of the matrices will have the same

dimensions as the original matrices.

4 3 1 8 1 5 (4 8) ( 3 1) (1 5) 12 4 4

Perform the indicated operation, if possible.

Example 3: 3 0 12 5 9 1

5 8 6 0 13 7

Size = _____ X _____ Size = _____ X _____ Are they the same size? _____

Example 4:

0 19 2 8

4 56 1 0

7 3

Size = _____ X _____ Size = _____ X _____ Are they the same size? _____

Reflection: 3

Scalar Multiplication

Ex: Find the indicated matrix. Let

6 1

0 2

3 4

2 5

A . Find -3A.

Multiply every entry in the matrix by the scalar.

6 1 18 3

0 2 0 63

3 4 9 12

2 5 6 15

Find the indicated matrix.

Example 5: Let 1 7 2 0B . Find -6B.

Example 6: Let

9

27

3

C . Find 1

3C

Example 7: Let 4 2

1 3A and

2 0

7 4B . Find A + 4B.

Two matrices are equal if their dimensions are the same, and all of the corresponding entries are equal.

Example 8: Determine the values of x and y that make the equation true.

4 2 5 1 9 13

8 1 5 2 12 3

x

y

Reflection: 4

Sample CCSD Common Exam Practice Question(s):

1. The tables show the number of students in band and choir by grade and gender.

Students in Band Students in Choir

Grade Girls Boys Grade Girls Boys

10 23 27 10 26 20

11 18 20 11 20 16

12 13 17 12 12 15

Which could be used to find the total number of girls and boys in the band and choir by grade level?

A.

23 27 26 20

18 20 20 16

13 17 12 15

B.

23 2726 20 12

18 2020 16 15

13 17

C.

23 26 20 27

18 20 16 20

13 12 15 17

D.

23 2627 20 17

18 2020 16 15

13 12

2. Find the values of x and y that make the equation true.

2 2 4 8 16 6

3 6 10 7 1

x

y

A. x = –8, y = –3

B. x = –7, y = 3

C. x = –6, y = –13

D. x = –1, y = –1

Reflection: 5

Section 4.2 Matrix Multiplication Objective(s): Simplify matrix expressions using the properties of matrices.

Essential Question: Is matrix multiplication commutative? Explain your answer.


Notes:

Matrix Multiplication:

In order to multiply matrices, the number of columns in the first matrix must equal the number of rows

in the second matrix. The product matrix will have the same number of rows of the first matrix and the

same number of columns of the second matrix.

Let A be an m n matrix, and B be an n p matrix. The product A B exists, and will have the

dimensions m p .

Determine if the matrices can be multiplied.

Example 1: Let A be a 3 4 matrix and let B be a 4 7 matrix.

Possible: yes or no Dimensions of product: _____ X _____





Ex: Find the product AB (if possible).

2 3

1 5

0 2

A 4 1

3 6B

Step One: Determine if the product exists. If it does, find its dimensions.

Matrix A is a 3 2 matrix. Matrix B is a 2 2 matrix. The number of columns in matrix

A equals the number of rows in matrix B. Therefore, the product exists and will be a

3 2 matrix.

Reflection: 6

Step Two: Multiply each entry in the rows of matrix A to each entry in the columns of matrix B.

Then find the sum of these products.

2 3

1 5

0 2

4 1

3 6 =

2 4 3 3 1

2 3

1 5

0 2

4 1

3 6 =

1 2 1 3 6 1 20

2 3

1 5

0 2

4 1

3 6 =

1 20 1 20

1(4) 5( 3) 19

2 3

1 5

0 2

4 1

3 6 =

1 20 1 20

19 1(1) 5(6) 19 29

2 3

1 5

0 2

4 1

3 6 =

1 20 1 20

19 29 19 29

0(4) ( 2)( 3) 6

2 3

1 5

0 2

4 1

3 6 =

1 20 1 20

19 29 19 29

6 0 1 2 6 6 12

Reflection: 7

Example 4: What is the product AB ?

1 3

4 2A ,

2 0

1 2B


AB


1 3

3 4A ,

0 2 4

1 3 2B


AB


1 3 2

2 0 4A ,

3 0

2 1

0 4

B


AB


7

2A ,

7 12

3 5

8 4

B


Reflection: 8


8 2 5A ,

7

8

6

B


AB


1. If

6 2

1 4

0 5

A and 4 2 5

4 6 1B , which is the product AB?

A. 26 25

18 37

B. 22 9

30 21

C.

16 24 28

20 22 20

20 30 5

D.

32 0 32

12 26 1

20 30 5

Reflection: 9

Section 4.3 Determinants and Inverse Matrices Objective(s): Find the determinant of a matrix. Find inverse matrices.

Essential Question: What is the special relationship between a matrix and its inverse?


Notes:

Determinant of a Matrix

Notation: Determinant of Matrix A = det A = A

Evaluating the Determinant of a 2 X 2 Matrix

a b

ad bcc d

Ex: What is the determinant of the matrix A? 3 1

7 9A

det 3 9 1 7 27 7 20A

Evaluate the determinant.

Example 1: What is the determinant of 9 3

7 6?


9 6?

Reflection: 10


8 3?

Example 4: What is the determinant of

1 1

10 2

2 10

?


x y?

Identity Matrix

The identity matrices are square matrices with ones on the main diagonal (from upper left to lower

right) and zeros everywhere else.

The identity matrix of a 2 X 2 2

1 0

0 1I


1 0 0

0 1 0

0 0 1

I


1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

I

Finding the Inverse of a 2 X 2.

If you have a matrix A, then A-1 (read A inverse) is found by using the following formula.

Ifa b

Ac d

, then 1 1 d b

Ac aad bc

Note: 0ad bc

Reflection: 11

Ex: Find the inverse of 3 5

2 4A .

14 51

2 34 3 5 2

4 51

2 312 10

4 51

2 32

1 14 5

2 2

1 12 3

2 2

522

312

A

Find the inverse, if it exists, of the given matrix.

Example 6: 3 5

6 5A Example 7:

5 0

5 4A

Example 8: 2 1

12 6A Example 9:

3 1

5 6A

Reflection: 12

Example 10: 2 1

3 8A Example 11:

0 4

0 7A


1. What is the determinant of 2 1

3 2?

A. –7

B. –2

C. –1

D. 1

Reflection: 13

Section 4.4 Solving Systems Using Inverse Matrices Objective(s): Solve systems of equations using matrices.

Essential Question: What are some pros and cons for solving systems using inverse matrices as opposed

to the substitution method or by linear combinations?


Notes:

Find the product of the two matrices.

Example 1: 2 4

6 3

x

y Example 2:

2 6

4 3

x

y

Write the system using matrices.

Example 3: 12x – 7y = 9 Example 4: -5x + 2y = 16

-2x + 6y = 4 13x – 8y = 5

Find the inverse of the coefficient matrix.

Example 5: 5x + 24y = 2

x + 5y = -3

A =

A-1 =

Reflection: 14

Solving Systems of Equations Using Inverse Matrices

1. Rewrite the system using matrices.

2. Find the inverse of the coefficient matrix, A-1.

3. Multiply the inverse matrix, A-1, times the constant matrix.

Ex: Solve the system 2 7 3

3 8 23

x y

x y using an inverse matrix.

Step One: Rewrite the system of equations as a matrices, where A is the coefficient matrix, X is

the variable matrix, and B is the matrix of constants.

2 7 3

3 8 23

x

y Note: Use matrix multiplication to show that this represents the original

system.

Step Two: Find the inverse of the coefficient matrix. 2 7

3 8A

18 71

3 22 8 7 3

8 71

3 237

A

Hint: Do not multiply by the scalar yet.

Step Three: Multiply the constant matrix by 1A .

8 7 31

3 2 2337

8 3 7 231

3 3 2 2337

1851

3737

5

1

x

y

Therefore, x = -5 and y = 1.

Solution: (-5, 1)

Reflection: 15

Solve the matrix equation.

Example 6: 2 3 1

1 1 3

x

y

A =

A-1 =

Multiply A-1B

x

y

Solution: ( , )

Example 7: 1 1 8

2 5 31

x

y

A =

A-1 =

Multiply A-1B

x

y

Solution: ( , )

Reflection: 16

Example 8: 5 1 3

1 3 23

x

y

A =

A-1 =

Multiply A-1B

x

y

Solution: ( , )

Example 9: 2 1 13

5 2 1

x

y

A =

A-1 =

Multiply A-1B

x

y

Solution: ( , )

Reflection: 17

Solve the system of equations using inverse matrices.

Example 10: 1

2 3 12

x y

x y

Example 11: 5 0

4

x y

x y

Example 12: 2 5

2

x y

x y

Reflection: 18


1. The flower shop in a grocery store sells flowers individually. The relationship between r, the

cost of one rose, and c, the cost of one carnation, is represented by the matrix equation below.

3 2 8

2 5 9

r

c

What is the cost of buying one rose?

A. $0.82

B. $1.00

C. $1.55

D. $2.00

2. Given the system of linear equations:

2 3

4 5 0

x y

x y

Which equation below shows the solution to the system using inverse matrices?

A.

12 1 3

4 5 0

x

y

B.

13 2 1

0 4 5

x

y

C.

12 1

3 04 5

x

y

D. 31

2 1 0

4 5

x

y

section 4.1 matrix operations - edl · section 4.1 matrix operations objective(s): organize data...

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