Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models

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71. (3.5) Given and ﬁnd

fg, and

72. (4.2) Given is a zero of

ﬁnd all zeroes of h, real and complex.

73. (5.6) Since 2005, cable installations for an Internet

company have been modeled by the function

, where C(t) represents cable

installations in thousands, t yr after 2005. In what

year will the number of installations be greater

than 30,000?

C1t2 15 ln 1t  12

h1x2 x

 x

 12,x  2

.f  g, f  g,

g1x2 x  2,f 1x2 x

 8

74. (5.6) If a set amount of money p is deposited

regularly (daily, weekly, monthly, etc.) n times per

year at a ﬁxed interest rate r, the amount of money A

accumulated in t years is given by the formula

shown. If a parent deposited $250 per month for

18 yr at 4.6% beginning when her ﬁrst child was

born, how much has been accumulated to help pay

for college expenses?

A 

pca1 

 1d

650 CHAPTER 7 Matrices and Matrix Applications 7–14

College Algebra G&M—

䊳

EXTENDING THE CONCEPT

69. Given the drawing shown, use a system of

equations and the matrix method to ﬁnd the

measure of the angles labeled as x and y. Recall

that vertical angles are equal and that the sum of

the angles in a triangle is 180°.

70. The system given here has a solution of

Find the value of a and b.

1 ab 1

2b 2a 513

2a 73b 8

11, 2, 32.

(x  59)

71

䊳

MAINTAINING YOUR SKILLS

Matrices serve a much wider purpose than just a convenient method for solving sys-

tems. To understand their broader application, we need to know more about matrix

theory, the various ways matrices can be combined, and some of their more practical

uses. The common operations of addition, subtraction, multiplication, and division are

all deﬁned for matrices, as are other operations. Practical applications of matrix theory

can be found in the social sciences, inventory management, genetics, operations

research, engineering, and many other ﬁelds.

A. Equality of Matrices

To effectively study matrix algebra, we ﬁrst give matrices a more general deﬁnition.

For the general matrix A, all entries will be denoted using the lowercase letter “a,” with

their position in the matrix designated by the dual subscript a

. The letter “i” gives the

row and the letter “j” gives the column of the entry’s location. The general

matrix A is written

m  n

LEARNING OBJECTIVES

In Section 7.2 you will see

how we can:

A. Determine if two matrices

are equal

B. Add and subtract

matrices

C. Compute the product

of two matrices

7.2 The Algebra of Matrices

cob19545_ch07_637-650.qxd 10/21/10 8:41 AM Page 650

A. You’ve just seen how

we can determine if two

matrices are equal

College Algebra G&M—

The size of a matrix is also referred to as its order, and we say the order of general

matrix A is Note that diagonal entries have the same row and column number,

, where Also, where the general entry of matrix A is a

, the general entry of

matrix B is b

, of matrix C is c

, and so on.

EXAMPLE 1

䊳

Identifying the Order and Entries of a Matrix

State the order of each matrix and name the entries corresponding to a

, a

; b

; and c

, c

a. b. c.

Solution

䊳

a. matrix A: order Entry (the row 2, column 2 entry is ).

There is no a

entry (A is only ).

b. matrix B: order Entry , entry .

c. matrix C: order Entry , entry .

Now try Exercises 7 through 12

䊳

Equality of Matrices

Two matrices are equal if they have the same order and their corresponding entries are

equal. In symbols, this means that if for all i and j.

EXAMPLE 2

䊳

Determining If Two Matrices Are Equal

Determine whether the following statements are true, false, or conditional. If false,

explain why. If conditional, ﬁnd values that will make the statement true.

a. b.

Solution

䊳

a. is false. The matrices have the same order and

entries, but corresponding entries are not equal.

b. is false. Their orders are not equal.

c. is conditional. The statement is true when

and is false otherwise.

Now try Exercises 13 through 16

䊳

2b ⫽ 4 1b ⫽ 22, c ⫽⫺2,a ⫺ 2 ⫽ 1 1a ⫽ 32,

⫽ c

a ⫺ 22b

c ⫺3

⫺2 ⫺3

⫽ c

3 ⫺21

5 ⫺43

d£

3 ⫺2

⫺43

⫽ c

⫺3 ⫺2

⫺2 ⫺3

d⫽ c

a ⫺ 22b

c ⫺3

⫽ c

3 ⫺21

5 ⫺43

d£

3 ⫺2

⫺43

§c

⫺2 ⫺3

d⫽ c

⫺3 ⫺2

⫽ b

A ⫽ B

⫽ 2.1c

⫽ 0.33 ⫻ 3.

⫽⫺4b

⫽ 53 ⫻ 2.

2 ⫻ 2

⫺3a

⫽⫺32 ⫻ 2.

C ⫽ £

0.2 ⫺0.5 0.7

⫺1 0.3 1

2.1 ⫺0.1 0.6

§B ⫽ £

3 ⫺2

1 5

⫺43

§A ⫽ c

⫺2 ⫺3

i ⫽ j.

m ⫻ n.

row m S

row i S

row 3 S

row 2 S

row 1 S

col 1 col 2 col 3 col j col n

ooo o o

is a general

matrix element

{

}

7–15 Section 7.2 The Algebra of Matrices 651

cob19545_ch07_651-663.qxd 10/21/10 9:47 PM Page 651

College Algebra G&M—

B. Addition and Subtraction of Matrices

A sum or difference of matrices is found by combining the corresponding entries. This

limits the operations to matrices of like orders, so that every entry in one matrix has a

“corresponding entry” in the other. This also means the result is a new matrix of like

order, whose entries are the corresponding sums or differences. If you attempt to add

matrices of unlike size on a graphing calculator, an error message is displayed: ERR:

DIM MISMATCH. Note that since a

represents a general entry of matrix A, [a

] rep-

resents the entire matrix.

Addition and Subtraction of Matrices

Given matrices A, B, and C having like orders.

The sum The difference

where where

EXAMPLE 3

䊳

Adding and Subtracting Matrices

Compute the sum or difference of the matrices indicated.

a. b. c.

Solution

䊳

a. sum of

and

add corresponding entries

difference of

and

subtract corresponding entries

Now try Exercises 17 through 20

䊳

Operations on matrices can be very laborious for larger matrices and for matrices with

noninteger or large entries. For these, we can turn to available technology for

assistance. This shifts our focus from a meticulous computation of entries, to carefully

entering each matrix into the calculator, double-checking each entry, and appraising

results to see if they’re reasonable.

⫽ £

1 ⫺8

⫺56

§ ⫽ £

3 ⫺ 2 ⫺2 ⫺ 6

1 ⫺ 15⫺ 0

⫺4 ⫺ 13⫺ 1⫺32

⫺ £

1 ⫺3

§C ⫺ A ⫽ £

3 ⫺2

⫺43

⫺32⫺1

⫺54 3

dA ⫹ B ⫽ £

1 ⫺3

§⫹

⫽ £

⫺30

§ ⫽ £

2 ⫹ 36⫹ 1⫺22

1 ⫹ 10⫹ 5

1 ⫹ 1⫺42 ⫺3 ⫹ 3

3 ⫺2

⫺43

§ A ⫹ C ⫽ £

1 ⫺3

§⫹

C ⫺ AA ⫹ BA ⫹ C

C ⫽ £

3 ⫺2

⫺43

§B ⫽ c

⫺32⫺1

⫺54 3

dA ⫽ £

1 ⫺3

⫺ b

4⫽ 3c

4.3a

⫹ b

4⫽ 3c

A ⫺ B ⫽ C,A ⫹ B ⫽ C,

652 CHAPTER 7 Matrices and Matrix Applications 7–16

Addition and subtraction are not

deﬁned for matrices of unlike order.

cob19545_ch07_651-663.qxd 11/29/10 9:36 AM Page 652

B. You’ve just seen how

we can add and subtract

matrices

EXAMPLE 4

䊳

Using Technology for Matrix Operations

Use a calculator to compute the difference for the matrices given.

Solution

䊳

The entries for matrix A are shown in Figure 7.9. After entering matrix B, exit to

the home screen [ (QUIT)], call up matrix A, press the (subtract)

key, then call up matrix B and press . The calculator quickly ﬁnds the difference

and displays the results shown in Figure 7.10. The last line on the screen shows the

result can be stored for future use in a new matrix C by pressing the key,

calling up matrix C, and pressing .

Now try Exercises 21 through 24

䊳

In Figure 7.10 the dots to the right on the calculator screen indicate there are addi-

tional digits or matrix columns that can’t ﬁt on the display, as often happens with larger

matrices or decimal numbers. Sometimes, converting entries to fraction form will pro-

vide a display that’s easier to read. Here, this is done by calling up the matrix C, and

using the 1:

䊳

Frac option. After pressing , all entries are converted to frac-

tions in simplest form (where possible), as in Figure 7.11. The third column can be

viewed by pressing the right arrow.

Since the addition of two matrices is deﬁned as the sum of corresponding entries,

we ﬁnd the properties of matrix addition closely resemble those of real number addi-

tion. Similar to standard algebraic properties, represents the product and

any subtraction can be rewritten as an algebraic sum: As noted in

the properties box, for any matrix A, the sum will yield the zero matrix Z,

a matrix of like size whose entries are all zeroes. Also note that matrix is the

additive inverse for A, while Z is the additive identity.

Properties of Matrix Addition

Given matrices A, B, C, and Z are matrices, with Z the zero matrix. Then,

I. matrix addition is commutative

II. matrix addition is associative

III. Z is the additive identity

IV. is the additive inverse of A

C. Matrices and Multiplication

The algebraic terms 2a and ab have counterparts in matrix algebra. The product 2A

represents a constant times a matrix and is called scalar multiplication. The product

AB represents the product of two matrices.

⫺AA ⫹ 1⫺A2⫽ 1⫺A2⫹ A ⫽ Z

A ⫹ Z ⫽ Z ⫹ A ⫽ A

1A ⫹ B2⫹ C ⫽ A ⫹ 1B ⫹ C2

A ⫹ B ⫽ B ⫹ A

m ⫻ n

⫺A

A ⫹ 1⫺A2

A ⫺ B ⫽ A ⫹ 1⫺B2.

⫺1

A⫺A

ENTER

MATH

Figure 7.9 Figure 7.10

ENTER

STO

ENTER

ⴚ

MODE

2nd

B ⫽ £

⫺7

0.75

0 ⫺5

⫺4

⫺5

§A ⫽ £

⫺0.5

0.9

⫺4

06⫺

A ⫺ B

7–17 Section 7.2 The Algebra of Matrices 653

College Algebra G&M—

Figure 7.11

cob19545_ch07_651-663.qxd 11/29/10 9:34 AM Page 653

Scalar Multiplication

Scalar multiplication is deﬁned by taking the product of the constant with each entry

in the matrix, forming a new matrix of like size. In symbols, for any real number k and

matrix A,

EXAMPLE 5

䊳

Computing Operations on Matrices

Given and compute the following:

a. b.

Solution

䊳

rewrite using algebraic addition

simplify

result

Now try Exercises 25 through 28

䊳

Matrix Multiplication

Consider a cable company offering three different levels of Internet service: Bronze—

fast, Silver—very fast, and Gold—lightning fast. Table 7.1 shows the number and

types of programs sold to households and businesses for the week. Each program has

an incentive package consisting of a rebate and a certain number of free weeks, as

shown in Table 7.2.

⫺

⫺11

⫺2 ⫺7

2 11.8

§⫽ £

⫺16 ⫹ 1⫺

2 ⫺12 ⫹ 1

⫺2 ⫹ 0 ⫺4 ⫹ 1⫺32

0 ⫹ 212⫹ 1⫺0.22

§⫽

⫺

0 ⫺3

2 ⫺0.2

§⫽ £

⫺16 ⫺12

⫺2 ⫺4

012

§⫹

1⫺

21321⫺

21⫺22

1⫺

21021⫺

2162

1⫺

21⫺421⫺

210.42

§⫽ £

1⫺421421⫺42132

1⫺421

21⫺42112

1⫺421021⫺421⫺32

§⫹

⫺4A⫺

B ⫽⫺4A ⫹ a⫺

b B

⫺1

⫺2 0.2

§⫽ £

21321

21⫺22

21021

2162

21⫺421

210.42

§⫽

B ⫽ a

b£

3 ⫺2

⫺4 0.4

⫺4A ⫺

B ⫽ £

3 ⫺2

⫺4 0.4

§,A ⫽ £

0 ⫺3

kA ⫽ 3ka

654 CHAPTER 7 Matrices and Matrix Applications 7–18

College Algebra G&M—

Table 7.1 Matrix A

Table 7.2 Matrix B

Bronze Silver Gold

Homes 40 20 25

Businesses 10 15 45

Rebate Free Weeks

Bronze $15 2

Silver $25 4

Gold $35 6

To compute the amount of rebate money the cable company paid to households

for the week, we would take the ﬁrst row (R1) in Table 7.1 and multiply by the

cob19545_ch07_651-663.qxd 10/21/10 9:47 PM Page 654

corresponding entries (bronze with bronze, silver with silver, and so on) in the ﬁrst

column (C1) of Table 7.2 and add these products. In matrix form, we have

Using R1 of Table 7.1

with C2 from Table 7.2 gives the number of free weeks awarded to homes:

Using the second row (R2) of

Table 7.1 with the two columns from Table 7.2 will give the amount of rebate money

and the number of free weeks, respectively, awarded to business customers. When all

computations are complete, the result is a product matrix P with order This is

because the product of R1 from matrix A, with C1 from matrix B, gives the entry in

position P

of the product matrix: .

Likewise, the product will give entry P

(310), the product of R2 with C1 will

give P

(2100), and so on. This “row column” multiplication can be generalized,

and leads to the following. Given matrix A and matrix B,s ⫻ tm ⫻ n

⫻

⫽ c

1975 310

2100 350

d£

15 2

25 4

35 6

§c

40 20 25

10 15 45

2 ⫻ 2.

340 20 254

§⫽ 40

2 ⫹ 20

4 ⫹ 25

6 ⫽ 310.

35 ⫽ $1975.340 20 254

§⫽ 40

15 ⫹ 20

25 ⫹

7–19 Section 7.2 The Algebra of Matrices 655

College Algebra G&M—

AB AB

cc c c

1s ⫻ t 21m ⫻ n21s ⫻ t 21m ⫻ n2

result will be an

matrixm ⫻ t

matrix multiplication is

possible only when

n ⫽ s

In less formal terms, matrix multiplication involves multiplying the row entries of

the ﬁrst matrix with the corresponding column entries of the second, and adding them

together. In Example 6, two of the matrix products [parts (a) and (b)] are shown in full

detail, with the ﬁrst entry of the product matrix color-coded.

EXAMPLE 6

䊳

Multiplying Matrices

Given the matrices A through E shown here, compute the following products:

a. AB b. CD c. DC d. AE e. EA

E ⫽ £

⫺2 ⫺1

§D ⫽ £

25 1

4 ⫺11

03⫺2

§C ⫽ £

⫺21 3

102

41⫺1

§B ⫽ c

dA ⫽ c

⫺21

In more formal terms, we have the following deﬁnition of matrix multiplication.

Matrix Multiplication

Given the matrix and the matrix If then

matrix multiplication is possible and the product AB is an matrix

where p

is product of the ith row of A with the jth column of B.

P ⫽ 3p

4,m ⫻ t

n ⫽ s,B ⫽ 3b

4.s ⫻ tA ⫽ 3a

4m ⫻ n

cob19545_ch07_651-663.qxd 10/21/10 9:47 PM Page 655

656 CHAPTER 7 Matrices and Matrix Applications 7–20

College Algebra G&M—

CD CD

(3 ⫻ 3) (3 ⫻ 3) (3 ⫻ 3) (3 ⫻ 3)

cc c c

result will be a

3 ⫻ 3 matrix

multiplication is possible

since 3 ⫽ 3

DC DC

(3 ⫻ 3) (3 ⫻ 3) (3 ⫻ 3) (3 ⫻ 3)

cc c c

result will be a

3 ⫻ 3 matrix

multiplication is possible

since 3 ⫽ 3

EA EA

(3 ⫻ 2) (2 ⫻ 2) (3 ⫻ 2) (2 ⫻ 2)

cc c c

result will be a

3 ⫻ 2 matrix

multiplication is possible

since 2 ⫽ 2

(2 ⫻ 2) (3 ⫻ 2)

multiplication is not possible since 2 ⫽ 3

AB AB

(4 ⫻ 3) (3 ⫻ 3) (4 ⫻ 3) (3 ⫻ 3)

cc c c

result will be a

4 ⫻ 3 matrix

multiplication is possible

since 3 ⫽ 3

Solution

䊳

Computation:

Now try Exercises 29 through 40

䊳

Example 6 shows that in general, matrix multiplication is not commutative. Parts

(b) and (c) show since we get different results, and parts (d) and (e) show

since AE is not deﬁned while EA is.

As with the addition and subtraction of matrices, matrix multiplication becomes

cumbersome and time consuming for larger matrices, and we will often turn to the

technology available in such cases.

EXAMPLE 7

䊳

Using Technology for Matrix Operations

Use a calculator to compute the product AB.

Solution

䊳

Carefully enter matrices A and B into the calculator,

then press (QUIT) to get to the home

screen. Use [A][B] , and the calculator ﬁnds the

product shown in the ﬁgure. Just for “fun,” we’ll

ENTER

MODE2nd

B ⫽ £

⫺0.7 1

0.5 3.2 ⫺3

⫺2

§A ⫽ ≥

2 ⫺30

⫺15 4

602

32⫺1

AE ⫽ EA,

CD ⫽ DC

EA ⫽ £

⫺2 ⫺1

§c

⫺21

d⫽ £

1 ⫺6

⫺63

AE ⫽ c

⫺21

d£

⫺2 ⫺1

DC ⫽ £

25 1

4 ⫺11

03⫺2

§£

⫺21 3

102

41⫺1

§⫽ £

5315

⫺55 9

⫺5 ⫺28

1⫺22122⫹ 112142⫹ (3)(0) 1⫺22152⫹ 1121⫺12⫹ (3)132 1⫺22112⫹ 112112⫹ (3)1⫺22

112122⫹ 102142⫹ 122(0) 112152⫹ 1021⫺12⫹ 122132112112⫹ 102112⫹ 1221⫺22

142122⫹ 112142⫹ 1⫺12(0) 142152⫹ 1121⫺12⫹ 1⫺12132142112⫹ 112112⫹ 1⫺121⫺22

CD ⫽ £

⫺2 1 3

102

41⫺1

§£

2 51

4 ⫺11

3 ⫺2

§⫽ £

0 ⫺2 ⫺7

211⫺3

12 16 7

1⫺22142⫹ 112162 1⫺22132⫹ 112112

132142⫹ 142162 132132⫹ 142112

AB ⫽ c

⫺2 1

4 3

6 1

d⫽ c

⫺2 ⫺5

36 13

AB AB

(2 ⫻ 2) (2 ⫻ 2) (2 ⫻ 2) (2 ⫻ 2)

cc c c

result will be a

2 ⫻ 2 matrix

multiplication is possible

since 2 ⫽ 2

cob19545_ch07_651-663.qxd 12/24/10 9:01 PM Page 656

Table 7.3 Matrix A

A Age Group

Sex 18–19 20–21 22–23

Female 1000 1500 500

Male 2500 3000 2000

Table 7.4 Matrix B

B Likelihood of Joining

Age Group Army Navy Air Force Marines

18–19 0.42 0.28 0.17 0.13

20–21 0.38 0.26 0.27 0.09

22–23 0.33 0.25 0.35 0.07

7–21 Section 7.2 The Algebra of Matrices 657

College Algebra G&M—

double check the ﬁrst entry of the product matrix:

(2)(1/2) ⫹ (⫺3)(0.5) ⫹ (0)(⫺2) ⫽⫺0.5. ✓

Now try Exercises 41 through 52

䊳

Properties of Matrix Multiplication

Earlier, Example 6 demonstrated that matrix multiplication is not commutative. Here

is a group of properties that do hold for matrices. You are asked to check these proper-

ties in the Exercise Set using various matrices. See Exercises 53 through 56.

Properties of Matrix Multiplication

Given matrices A, B, and C for which the products are deﬁned:

I. matrix multiplication is associative

II. matrix multiplication is distributive from the left

III. matrix multiplication is distributive from the right

IV. a constant k can be distributed over addition

We close this section with an application of matrix multiplication. There are many

other interesting applications in the Exercise Set.

EXAMPLE 8

䊳

Using Matrix Multiplication to Track Volunteer Enlistments

In a certain country, the number of males and females that will join the military depends on their age.

This information is stored in matrix A (Table 7.3). The likelihood a volunteer will join a particular

branch of the military also depends on their age, with this information stored in matrix B (Table 7.4).

(a) Compute the product P ⫽ AB and discuss/interpret what is indicated by the entries P

, P

, and

of the product matrix. (b) How many males are expected to join the Navy this year?

k1A ⫹ B2⫽ kA ⫹ kB

1B ⫹ C2A ⫽ BA ⫹ CA

A1B ⫹ C2⫽ AB ⫹ AC

A1BC2⫽ 1AB2C

⫺0.7 1

0.5 3.2 ⫺3

⫺2

§AB ⫽ ≥

2 ⫺30

⫺15 4

602

32⫺1

Solution

䊳

a. Matrix A has order and matrix B has order The product matrix P can be found

and is a matrix. Carefully enter the matrices in your calculator. Figure 7.12 shows the

entries of matrix B. Using , the calculator ﬁnds the product matrix shown in

Figure 7.13. Pressing the right arrow shows the complete product matrix is

The entry P

is the product of R1 from A and C1 from B, and indicates that for the year, 1155

females are projected to join the Army. In like manner, entry P

shows that 750 females are

projected to join the Air Force. Entry P

indicates that 735 males are projected to join the

Marines.

P ⫽ c

1155 795 750 300

2850 1980 1935 735

ENTER

3A43B4

2 ⫻ 4

3 ⫻ 4.2 ⫻ 3

cob19545_ch07_651-663.qxd 11/29/10 9:59 AM Page 657

658 CHAPTER 7 Matrices and Matrix Applications 7–22

College Algebra G&M—

Figure 7.12

Figure 7.13

C. You’ve just seen how

we can compute the product

of two matrices

b. The product R2 (males) (Navy) gives meaning 1980 males are expected to

join the Navy.

Now try Exercise 59 through 66

䊳

⫽ 1980,

䊳

CONCEPTS AND VOCABULARY

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.

7.2 EXERCISES

2. The sum of two matrices (of like size) is found by

adding the corresponding entries. In symbols,

A ⫹ B ⫽

1. Two matrices are equal if they are like size and the

corresponding entries are equal. In symbols,

if .

⫽

A ⫽ B

3. The product of a constant times a matrix is called

multiplication.

4. The size of a matrix is also referred to as its .

The order of is .

A ⫽ c

123

456

5. Give two reasons why matrix multiplication is

generally not commutative. Include several

examples using matrices of various sizes.

6. Discuss the conditions under which matrix

multiplication is deﬁned. Include several examples

using matrices of various sizes.

䊳

DEVELOPING YOUR SKILLS

State the order of each matrix and name the entries in

positions a

and a

if they exist. Then name the

position a

of the “5” in each.

7. 8.

9. 10.

11. 12. £

89 55 34 21

13 8 5 3

2110

§£

⫺21⫺7

081

5 ⫺14

2 0.4

⫺0.1 5

0.3 ⫺3

§c

2 ⫺3 0.5

0 5 6

⫺11

§c

1 ⫺3

5 ⫺7

Determine if the following statements are true, false, or

conditional. If false, explain why. If conditional, ﬁnd values

of a, b, c, p, q, and r that will make the statement true.

13.

14.

15.

16. £

2p ⫹ 1 ⫺59

1120

q ⫹ 59⫺2r

§⫽ £

7 ⫺52⫺ q

13r 0

⫺23p ⫺8

⫺23 a

2b ⫺54

0 ⫺93c

§⫽ £

c 3 ⫺4

6 ⫺5 ⫺a

0 ⫺3b ⫺6

≥

⫺7

⫺1

⫺2

¥ ⫽ c

1.5 ⫺1.4 1.3

⫺0.5 ⫺0.4 0.3

14 18

116 132 164

d⫽ c

12212

4412 8

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7–23 Section 7.2 The Algebra of Matrices 659

College Algebra G&M—

For matrices A through J as given, perform the

indicated operation(s), if possible. If an operation

cannot be completed, state why. Use a calculator only

for those exercises designated by an icon.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. ED 30. DE

31. AH 32. HA

33. CB 34. FH

35. HF 36. EB

37. H

38. F

39. FE 40. EF

F ⫺

E ⫺ 3D

2E ⫹ 3G3H ⫺ 2A

I ⫺ GG ⫹ I

A ⫺ JH ⫹ J

G ⫹ DF ⫹ H

E ⫹ GA ⫹ H

J ⫽ ≥

⫺

¥I ⫽

⫺

H ⫽ c

8 ⫺3

⫺52

dG ⫽ £

⫺12 0

01⫺2

⫺4 ⫺36

F ⫽ c

6 ⫺39

12 0 ⫺6

dE ⫽ £

1 ⫺20

0 ⫺12

43⫺6

D ⫽ £

100

010

001

§C ⫽ £

2 0.5

0.2 5

⫺13

B ⫽ £

⫺3

§A ⫽ c

For matrices A through H as given, use a calculator to

perform the indicated operation(s), if possible. If an

operation cannot be completed, state why.

41. AH 42. HA

43. EG 44. GE

45. HB 46. BH

47. DG 48. GD

49. C

50. E

51. FG 52. AF

For Exercises 53 through 56, use a calculator and

matrices A, B, and C to verify each statement.

53. Matrix multiplication is not generally commutative:

(a) (b) and (c)

54. Matrix multiplication is distributive from the left:

55. Matrix multiplication is distributive from the right:

56. Matrix multiplication is associative:

1AB2C ⫽ A1BC2.

1B ⫹ C2A ⫽ BA ⫹ CA.

A1B ⫹ C2⫽ AB ⫹ AC.

BC ⫽ CB.AC ⫽ CA,AB ⫽ BA,

C ⫽ £

45 ⫺13

⫺610⫺15

21 ⫺28 36

B ⫽ £

0.3 ⫺0.4 1.2

⫺2.5 2 0.9

1 ⫺0.5 0.2

§A ⫽ £

⫺13 5

27⫺1

406

H ⫽ ≥

⫺3

¥G ⫽

⫺1

F ⫽ c

⫺8

dE ⫽ £

1 ⫺20

0 ⫺12

43⫺6

D ⫽ £

100

010

001

§C ⫽ £

213

B ⫽ c

dA ⫽ c

⫺54

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