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5 Matrix Operations: Getting Information from Data

# 5 Matrix Operations: Getting Information from Data

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612

CHAPTER 7

Systems of Equations and Data in Categories

Solution

(a) Since A and B are matrices with the same dimension 13 * 22 , we can add

them.

2

A + B = £0

7

-3

1

5§ + £-3

- 12

2

0

3

1§ = £-3

2

9

0

6

d - c

5

8

-6

1

d = c

9

-8

-3

3

2

(b) Since C and D are matrices with the same dimension 12 * 32 , we can

subtract them.

C -D = c

7

0

-3

1

0

1

-3

0

6

d

-4

(c) C + A is not defined because we cannot add matrices with different dimensions.

(d) To find 5A, we multiply each entry in A by 5.

2

5A = 5 £ 0

7

2

-3

10

5§ = £ 0

- 12

35

- 15

25 §

- 52

NOW TRY EXERCISES 7 AND 15

■ Matrix Multiplication

In Section 7.4 we learned how to multiply a matrix times a column matrix. We now

describe how to use this to define matrix multiplication in general (when the second

factor has more than just one column).

First, the product AB of two matrices A and B is defined only when the number

of columns in A is equal to the number of rows in B. This means that if we write their

dimensions side by side, the two inner numbers must match.

Matrices

Dimensions

A

m *n

B

n *k

Columns in A

Rows in B

If the dimensions of A and B match in this way, then the product AB is a matrix of

dimension m * k. Before describing the procedure for obtaining the elements of AB,

we define the inner product of a row of A and a column of B.

b1

b

If 3a1 a2 p an 4 is a row of A and if ≥ 2 ¥ is a column of B, then their

o

bn

inner product is the number a1b1 + a2b2 + p + anbn. For example taking the inner

product, we have

32

-1

0

5

6

44 ≥

¥ = 2 # 5 + 1- 12 # 6 + 0 # 1- 32 + 4 # 2 = 12

-3

2

SECTION 7.5

Matrix Operations: Getting Information from Data

613

Matrix Multiplication

If A is an m * n matrix and B is an n * k matrix, then their product AB is an

m * k matrix C.

AB = C

The entries of the matrix C are calculated as follows: The entry in the ith row

and jth column of the matrix C is the inner product of the ith row of A and the

jth column of B.

The following diagram shows that the entry cij in the ith row and jth column of

the product matrix AB is obtained by multiplying the entries in the ith row of A by

the corresponding entries in the jth column of B and then adding the results.

jth column of B

ith row of A

cٗ ٗ ٗd

ٗ

ٗ

ٗ

£

Entry in ith row and

jth column of AB

§ = c

cij d

The next example illustrates the process.

example 2

Multiplying Matrices

Calculate, if possible, the products AB and BA.

A= c

1

-1

3

d

0

B = c

-1

0

5

4

2

d

7

Solution

Inner numbers match,

so product is defined.

Since matrix A has dimension 2 * 2 and matrix B has dimension 2 * 3, the product

AB is defined and has dimension 2 * 3. We can write

AB = c

2* 2 2* 3

Outer numbers give dimension

of product: 2 * 3.

1

-1

3 -1

dc

0

0

2

d = c

7

5

4

?

?

?

?

?

?

d

where the question marks must be filled in by using the rule defining the product of

two matrices. If we define C = AB = 3cij 4 , then the entry c11 is the inner product of

the first row of A and the first column of B.

c

1

-1

3 -1

dc

0

0

5

4

2

d

7

1 # 1- 12 + 3 # 0 = - 1

614

CHAPTER 7

Systems of Equations and Data in Categories

Similarly, we calculate the remaining entries of the product as follows.

Entry

Inner product of

Value

Product matrix

c12

c

1

-1

3 -1

dc

0

0

5

4

2

d

7

1 # 5 + 3 # 4 = 17

c

-1

17

c13

c

1

-1

3 -1

dc

0

0

5

4

2

d

7

1 # 2 + 3 # 7 = 23

c

-1

17

23

d

c21

c

1

-1

3 -1

dc

0

0

5

4

2

d

7

1- 12 # 1- 12 + 0 # 0 = 1

c

-1

1

17

23

d

c22

c

1

-1

3 -1

dc

0

0

5

4

2

d

7

1- 12 # 5 + 0 # 4 = - 5

c

-1

1

17

-5

23

d

c23

c

1

-1

3 -1

dc

0

0

5

4

2

d

7

1- 12 # 2 + 0 # 7 = - 2

c

-1

1

17

-5

23

d

-2

d

Thus we have

Not equal, so product

not defined.

AB = c

-1

1

17

-5

23

d

-2

The product BA is not defined, however, because the dimensions of B and A are

2 *3

2*3 2*2

and

2 *2

The inner two numbers are not the same, so the rows and columns won’t match up

when we try to calculate the product.

NOW TRY EXERCISES 11 AND 13

Graphing calculators are capable of performing all the matrix operations. In the

next example we use a graphing calculator to multiply two matrices.

e x a m p l e 3 Multiplying Matrices (Graphing Calculator)

Calculate the products AB and BA.

3

A = £ 2

-1

0

2

1

5

-4§

2

1

B = £0

3

-1

7

8

5

-2

Solution

Using a graphing calculator, we find the product AB.

Calculator output

[A] * [B]

[[18

37

5 ]

[-10 -20 30]

[5

24

-3]

Product AB

18

£ - 10

5

37

- 20

24

5

30 §

-3

SECTION 7.5

Matrix Operations: Getting Information from Data

615

Similarly, we find the product BA.

Calculator output

Product BA

[B] * [A]

[[-4 3

19 ]

[8

20 -16]

[27 14 -21]]

2

-4

£ 8

27

3

20

14

19

- 16 §

- 21

NOW TRY EXERCISE 19

■ Getting Information from Categorical Data

We will see in the next example how we can get information about categorical data

by multiplying matrices. (Compare to Example 5 in Section 7.4.)

example 4

Getting Information by Multiplying Matrices

Josh sells organic produce at an open-air market three days a week: Thursdays,

Fridays, and Saturdays. He sells oranges, broccoli, and beans. Matrix A tabulates the

number of pounds of produce that he obtains from his suppliers every week; he is always able to sell his complete supply by the end of each day. Matrix C gives the

prices per pound (in dollars) he charged on two different weeks. Find the product AC,

and interpret the entries of AC.

Produce

Oranges Broccoli

T

50

A = £ 70

45

T

20

35

15

Beans

T

10 d

30 § d

25 d

Thursday

Friday

Day

Saturday

Prices

Week 1 Week 2

T

0.90

C = £ 1.20

1.50

T

1.20 d

1.60 § d

1.90 d

Oranges

Broccoli Produce

Beans

Solution

We use a graphing calculator to find the product AC.

Calculator output

[A] * [C]

[[84

111

]

[150 197

]

[96

125.5]]

Product AC

84.00

£ 150.00

96.00

111.00

197.00 §

125.50

616

CHAPTER 7

Systems of Equations and Data in Categories

The product matrix gives us the revenue for each day of each week.

Week 1: A total of \$84.00 of produce was sold on Thursday, \$150.00 on

Friday, and \$96.00 on Saturday.

Week 2: A total of \$111.00 of produce was sold on Thursday, \$197.00 on

Friday, and \$125.50 on Saturday.

NOW TRY EXERCISE 27

7.5 Exercises

CONCEPTS

Fundamentals

1. We can add or (subtract) two matrices only if they have the same _______.

2. We can multiply two matrices only if the number of _______ in the first matrix is the

same as the number of _______ in the second matrix.

3. If A is a 3 * 3 matrix and B is a 2 * 3 matrix, which of the following matrix

multiplications are possible?

(a) AB

(b) BA

(c) AA

(d) BB

4. Fill in the missing entries in the product matrix.

3

£-1

1

1

2

3

-1

2

0§ £ 3

2

-2

3

-2

1

ٗ -7

-2

4

ٗ§

-1§ = £ 7 -7

ٗ -5 -5

0

5. Which of the following operations can we perform on a matrix A of any dimension?

(a) A + A

(b) 2A

(c) AA

6. If A is a matrix for which AA is defined, what must be true about the dimension of A?

SKILLS

7–14

7. c

2

-5

1

9. 3 £ 4

1

2

11. £ 1

2

1

13. c

-1

Perform the matrix operation, or explain why the operation is not defined.

6

-1

d + c

3

6

-3

d

2

2

-1§

0

6

1

3§ £ 3

4 -2

2 1

dc

4 2

8. c

0

1

1

10. 2 £ 1

0

-2

0

-2

2

12. c

3

d

-1

2

6

2

14. £ 0

1

1

2

d - c

0

1

1

1

1

0

1

1

3

0

1

1§ + £2

1

3

1

2

d£ 3

4

-2

-3

5

1§ c d

1

2

-1

d

-2

1

3

1

1

-2

0

SECTION 7.5

15–26

Matrix Operations: Getting Information from Data

617

Matrices A, B, C, D, E, F, G, and H are defined as follows. Perform the indicated

matrix operation(s), or explain why it cannot be performed.

A= c

2

0

D = 37

5

G= £ 6

-5

CONTEXTS

1

2

-5

d

7

B= c

34

1

E = £2§

0

-3

1

2

10

2

H= c

3

1

-1

5

d

3

C = c

2

0

1

F = £0

0

- 52

2

0

1

0

0

d

-3

0

1

1

d

-1

3

2

15. (a) B + C

(b) B + F

16. (a) C - B

(b) 2C - 6B

17. (a) 5A

(b) C - 5A

18. (a) 3B + 2C

(b) 2H + D

(b) DA

20. (a) DH

(b) HD

21. (a) AH

(b) HA

22. (a) BC

(b) BF

2

23. (a) GF

(b) GE

24. (a) B

25. (a) A2

(b) A3

26. (a) (DA)B

(b) F 2

(b) D(AB)

27. Fast-Food Sales A small fast-food chain with restaurants in Santa Monica, Long

Beach, and Anaheim sells only hamburgers, hot dogs, and milk shakes. On a certain

day, sales were distributed according to matrix A. The price of each item (in dollars) is

given by matrix C.

(a) Calculate the product matrix CA.

(b) What is the total profit from the Long Beach restaurant?

(c) What is the total profit (from all three restaurants)?

Number of items sold

Santa

Long

Monica

Beach

Anaheim

T

T

4000

A = £ 400

700

T

1000

300

500

3500 d Hamburgers

200 § d Hot dogs

Item

9000 d Milk shakes

Item

Hamburger Hot dog

T

C = 3 0.90

T

0.80

Milk shake

T

1.104 d Price

28. Car-Manufacturing Profits A specialty car manufacturer has plants in Auburn,

Biloxi, and Chattanooga. Three models are produced, with daily production given in

matrix A. Because of a wage increase, February profits are lower than January profits.

The profit (in dollars) per car is tabulated by model in matrix B.

(a) Calculate the product AB.

(b) Assuming that all cars produced were sold, what was the daily profit in January

from the Biloxi plant?

(c) What was the total daily profit (from all three plants) in February?

618

CHAPTER 7

Systems of Equations and Data in Categories

Cars produced each day

Model K

Model R

Model W

T

T

12

A= £ 4

8

10

4

9

T

0 d Auburn

City

20 § d Biloxi

12 d Chattanooga

January February

1000

B = £ 2000

1500

500 d Model K

1200 § d Model R

1000 d Model W

29. Canning Tomato Products Jaeger Foods produces tomato sauce and tomato paste,

canned in small, medium, large, and giant sized cans. The matrix A gives the size (in

ounces) of each container. The matrix B tabulates one day’s production of tomato sauce

and tomato paste.

(a) Calculate the product AB.

(b) Interpret the entries in the product matrix AB.

Size

Small Medium Large

T

A = 36

Giant

T

T

T

10

14

284

d Ounces

Cans of sauce Cans of paste

T

T

2000

3000

B= ≥

2500

1000

2500

1500

¥

1000

500

d

d

d

d

Small

Medium

Large

Giant

30. Produce Sales A farmer’s three children, Amy, Beth, and Chad, run three roadside

produce stands during the summer months. One weekend they all sell watermelons,

yellow squash, and tomatoes. Matrices A and B tabulate the number of pounds of each

product sold by each sibling on Saturday and Sunday. Matrix C gives the price per

pound (in dollars) for each type of produce that they sell. Perform each of the following

matrix operations, and interpret the entries in each result.

(a) AC

(b) BC

(c) A + B

(d) 1A + B2 C

Melons

Saturday

Squash

T

T

120

A = £ 40

60

50

25

30

Melons

Sunday

Squash

Tomatoes

T

T

T

60

20

25

30 d Amy

20 § d Beth

100

B = £ 35

60

Tomatoes

T

60 d Amy

30 § d Beth

Price per pound

0.10 d Melons

C = £ 0.50 § d Squash

1.00 d Tomatoes

SECTION 7.6

2

Matrix Equations: Solving a Linear System

619

7.6 Matrix Equations: Solving a Linear System

The Inverse of a Matrix

Matrix Equations

Modeling with Matrix Equations

IN THIS SECTION… we express a system of equations as a single matrix equation and

then solve the system by solving the matrix equation.

In the preceding section we saw that when the dimensions are appropriate, matrices

can be added, subtracted, and multiplied. In this section we investigate “division” of

matrices. With this operation we can solve equations that involve matrices.

2

■ The Inverse of a Matrix

First, we define identity matrices, which play the same role for matrix multiplication that the number 1 does for ordinary multiplication of numbers; that is,

1 # a = a # 1 = a for all numbers a. In the following definition the term main diagonal refers to the entries of a square matrix whose row and column numbers

are the same. These entries stretch diagonally down the matrix, from top left to

bottom right.

Identity Matrix

The identity matrix In is the n * n matrix for which each main diagonal

entry is a 1 and for which all other entries are 0.

Thus, the 2 * 2, 3 * 3, and 4 * 4 identity matrices are

I2 = c

1

0

0

d

1

1

0

I4 =

0

0

0

1

0

0

1

I3 = Ê 0

0

0

1

0

0

1

0

0

1

0

0

0

Ơ

0

1

Identity matrices behave like the number 1 in the sense that

A # In = A

and

B # In = B

whenever these products are defined.

The following matrix products show how multiplying a matrix by an identity

matrix of the appropriate dimension leaves the matrix unchanged.

c

1

0

-1

£ 12

-2

0

3

dc

1 -1

7

1

0

1

2

1

3§ £0

7 0

6

3

d = c

7

-1

5

2

0

1

0

0

-1

0 § = £ 12

1

-2

6

d

7

5

2

7

1

0

1

2

7

620

CHAPTER 7

Systems of Equations and Data in Categories

If A and B are n * n matrices, and if AB = BA = In, then we say that B is the

inverse of A, and we write B = A -1. The concept of the inverse of a matrix is analogous to that of the reciprocal of a real number.

Inverse of a Matrix

Let A be a square n * n matrix. If there exists an n * n matrix A -1 with the

property that

AA -1 = A -1A = In

then we say that A -1 is the inverse of A.

e x a m p l e 1 Verifying That a Matrix Is an Inverse

Verify that matrix B is the inverse of matrix A.

A = c

2

5

3

d

1

B = c

and

3 -1

d

-5

2

Solution

We perform the matrix multiplications to show that AB = I and BA = I.

c

2

5

c

3

-5

1

3

dc

3 -5

-1 2

dc

2 5

-1

2 # 3 + 11- 52

d = c #

2

5 3 + 31- 52

21- 12 + 1 # 2

1

d = c

#

51- 12 + 3 2

0

0

d

1

1

3 # 2 + 1- 125

d = c

3

1- 522 + 2 # 5

3 # 1 + 1- 123

1

d = c

#

1- 521 + 2 3

0

0

d

1

NOW TRY EXERCISE 3

The inverse of a matrix can be found by using a process that involves elementary row operations. This process is programmed into graphing calculators. On

TI-83 calculators, matrices are stored in memory by using names such as [A], [B],

[C], and so on. To find the inverse of [A], we key in

[A]

XϪ1

ENTER

e x a m p l e 2 Finding the Inverse of a Matrix

Find the inverse of matrix A.

1

A= £ 2

-3

-2

-3

6

-4

-6§

15

Solution

A graphing calculator gives the following output for the inverse of matrix A.

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