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

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660 CHAPTER 7 Matrices and Matrix Applications 7–24

College Algebra G&M—

䊳

WORKING WITH FORMULAS

The perimeter and area of a rectangle can be simultaneously calculated using the matrix formula shown, where L

represents the length and W represents the width of the rectangle. Use the matrix formula and your calculator to ﬁnd

the perimeter and area of the rectangles shown, then check the results using and

57. 58.

A ⫽ LW.P ⫽ 2L ⫹ 2W

W 0

dⴝ c

Perimeter

Area

6.374 cm

4.35 cm

5.02 cm

3.75 cm

䊳

APPLICATIONS

59. Custom T’s designs and sells specialty T-shirts and

sweatshirts, with factories in Verdi and Minsk. The

company offers this apparel in three quality levels:

standard, deluxe, and premium. Last fall the Verdi

plant produced 3820 standard, 2460 deluxe, and

1540 premium T-shirts, along with 1960 standard,

1240 deluxe, and 920 premium sweatshirts. The

Minsk plant produced 4220 standard, 2960 deluxe,

and 1640 premium T-shirts, along with 2960

standard, 3240 deluxe, and 820 premium

sweatshirts in the same time period.

a. Write a “production matrix” for each

plant Verdi, Minsk], with a T-shirt

column, a sweatshirt column, and three rows

showing how many of the different types of

apparel were manufactured.

b. Use the matrices from part (a) to determine

how many more or fewer articles of clothing

were produced by Minsk than Verdi.

c. Use scalar multiplication to ﬁnd how many

shirts of each type will be made at Verdi and

Minsk next fall, if each is expecting a 4%

increase in business.

d. Write a matrix that shows Custom T’s total

production next fall (from both plants), for

each type of apparel.

60. Terry’s Tire Store sells automobile and truck tires

through three retail outlets. Sales at the Cahokia store

for the months of January, February, and March

broke down as follows: 350, 420, and 530 auto tires

and 220, 180, and 140 truck tires. The Shady Oak

branch sold 430, 560, and 690 auto tires and 280,

320, and 220 truck tires during the same 3 months.

Sales ﬁgures for the downtown store were 864, 980,

and 1236 auto tires and 535, 542, and 332 truck tires.

a. Write a “sales matrix” for each store

Cahokia, Shady Oak,

Downtown], with January, February, and

D SS S3C S

2 ⫻ 3

M S3V S

3 ⫻ 2

March columns, and two rows showing the

sales of auto and truck tires respectively.

b. Use the matrices from part (a) to determine

how many more or fewer tires of each type the

downtown store sold (each month) over the

other two stores combined.

c. Market trends indicate that for the same three

months in the following year, the Cahokia

store will likely experience a 10% increase in

sales, the Shady Oak store a 3% decrease, with

sales at the downtown store remaining level

(no change). Write a matrix that shows the

combined monthly sales from all three stores

next year, for each type of tire.

61. Home improvements: Dream-Makers Home

Improvements specializes in replacement windows,

replacement doors, and new siding. During the

peak season, the number of contracts that came

from various parts of the city (North, South, East,

and West) are shown in matrix C. The average

proﬁt per contract is shown in matrix P. Compute

the product PC and discuss what each entry of the

product matrix represents.

N S E W

Windows Doors Siding

62. Classical music: Station 90.7—The Home of

Classical Music—is having their annual fund

drive. Being a loyal listener, Mitchell decides that

for the next 3 days he will donate money according

to his favorite composers, by the number of times

their music comes on the air: $3 for every piece by

Mozart (M), $2.50 for every piece by Beethoven

(B), and $2 for every piece by Vivaldi (V).

31500 500 25004⫽ P

Windows

Doors

Siding

9654

7576

2352

§⫽ C

cob19545_ch07_651-663.qxd 10/21/10 9:48 PM Page 660

7–25 Section 7.2 The Algebra of Matrices 661

College Algebra G&M—

This information is displayed in matrix D. The

number of pieces he heard from each composer is

displayed in matrix C. Compute the product DC

and discuss what each entry of the product matrix

represents.

Mon. Tue. Wed.

MBV

63. Pizza and salad: The science department and math

department of a local college are at a pre-semester

retreat, and decide to have pizza, salads, and soft

drinks for lunch. The quantity of food ordered by

each department is shown in matrix Q. The cost of

the food item at each restaurant is shown in matrix

C using the published prices from three popular

restaurants: Pizza Home (PH), Papa Jeff’s (PJ), and

Dynamos (D).

a. What is the total cost to the math department if

the food is ordered from Pizza Home?

b. What is the total cost to the science department

if the food is ordered from Papa Jeff’s?

c. Compute the product QC and discuss the

meaning of each entry in the product matrix.

Pizza Salad Drink

PH PJ D

64. Manufacturing pool tables: Cue Ball

Incorporated makes three types of pool tables, for

homes, commercial use, and professional use. The

amount of time required to pack, load, and install

each is summarized in matrix T, with all times in

hours. The cost of these components in dollars per

hour, is summarized in matrix C for two of its

warehouses, one on the west coast and the other in

the midwest.

a. What is the cost to package, load, and install a

commercial pool table from the coastal

warehouse?

b. What is the cost to package, load, and install a

commercial pool table from the warehouse in

the midwest?

c. Compute the product TC and discuss the

meaning of each entry in the product matrix.

Pizza

Salad

Drink

8 7.5 10

1.5 1.75 2

0.90 1 0.75

§⫽ C

Science

Math

8 12 20

10 8 18

d⫽ Q

33 2.5 24⫽ D

4 3 5

3 2 4

2 3 3

§⫽ C

Pack Load Install

Coast Midwest

65. Joining a club: Each school year, among the

students planning to join a club, the likelihood a

student joins a particular club depends on their

class standing. This information is stored in matrix

C. The number of males and females from each

class that are projected to join a club each year is

stored in matrix J. Compute the product JC and use

the result to answer the following:

a. Approximately how many females joined the

chess club?

b. Approximately how many males joined the

writing club?

c. What does the entry p

of the product matrix

tells us?

Fresh Soph Junior

Spanish Chess Writing

66. Designer shirts: The SweatShirt Shoppe sells

three types of designs on its products: stenciled (S),

embossed (E), and applique (A). The quantity of

each size sold is shown in matrix Q. The retail

price of each sweatshirt depends on its size and

whether it was ﬁnished by hand or machine. Retail

prices are shown in matrix C. Assuming all stock is

sold, compute the product QC and use the result to

answer the following.

a. How much revenue was generated by the large

sweatshirts?

b. How much revenue was generated by the

extra-large sweatshirts?

c. What does the entry p

of the product matrix

QC tell us?

S E A Hand Machine

40 25

60 40

90 60

§⫽ C

med

large

x-large

30 30 15

60 50 20

50 40 30

§⫽ Q

Fresh

Soph

Junior

0.6 0.1 0.3

0.5 0.2 0.3

0.4 0.2 0.4

§⫽ C

Female

Male

25 18 21

22 19 18

d⫽ J

Pack

Load

Install

10 8

12 10.5

13.5 12.5

§⫽ C

Home

Comm

Prof

1 0.2 1.5

1.5 0.5 2.2

1.75 0.75 2.5

§⫽ T

cob19545_ch07_651-663.qxd 11/29/10 10:01 AM Page 661

662 CHAPTER 7 Matrices and Matrix Applications 7–26

College Algebra G&M—

䊳

EXTENDING THE CONCEPT

67. For the matrix A shown, use your calculator to

compute A

, A

, and A

. Do you notice a

pattern? Try to write a “matrix formula” for A

where n is a positive integer, then use your formula

to ﬁnd A

. Check results using a calculator.

A ⫽ £

101

111

101

68. The matrix has some very

interesting properties. Compute the powers M

, M

, and M

, then discuss what you ﬁnd. Try to

ﬁnd/create another matrix that has similar

properties.

2 ⫻ 2

M ⫽ c

⫺3 ⫺2

䊳

MAINTAINING YOUR SKILLS

69. (6.2) Solve the system using elimination.

70. (2.1/4.6) Given f(x) ⫽ x

⫹ 10x

⫺ 9, solve

1x2ⱖ 0.

•

x ⫹ 2y ⫺ z ⫽ 3

⫺2x ⫺ y ⫹ 3z ⫽⫺5

5x ⫹ 3y ⫺ 2z ⫽ 2

71. (5.4) Evaluate using the change-of-base formula,

then check using exponentiation.

log

72. (4.1) Find the quotient using synthetic division,

then check using multiplication.

⫺ 9x ⫹ 10

x ⫺ 2

䊳

MID-CHAPTER CHECK

State the size of each matrix and identify the entry in

second row, third column.

Write each system in matrix form and solve using row

operations to triangularize the matrix. If the system is

linearly dependent, write the solution using a parameter.

3. 4.

6. For matrices A and B given, compute:

a. b. c. 5A ⫹ B

BA ⫺ B

B ⫽ c

10 15

⫺30 ⫺5

dA ⫽ c

⫺3 ⫺2

•

x ⫹ y ⫺ 3z ⫽⫺11

4x ⫺ y ⫺ 2z ⫽⫺4

3x ⫺ 2y ⫹ z ⫽ 7

•

⫺x ⫹ y ⫺ 5z ⫽ 23

2x ⫹ 4y ⫺ z ⫽ 9

3x ⫺ 5y ⫹ z ⫽ 1

2x ⫹ 3y ⫽⫺5

⫺5x ⫺ 4y ⫽ 2

B ⫽ c

⫺21

0 ⫺3

A ⫽ £

0.4 1.1 0.2

⫺0.2 0.1 ⫺0.9

0.7 0.4 0.8

7. For matrices C and D given, use a calculator to ﬁnd:

a. b. c. CD

8. For the matrices A, B, C, and D given, compute the

products indicated (if possible):

a. AC b. c. BA d.

9. Create a system of equations to model this exercise,

then write the system in matrix form and solve. The

campus bookstore offers both new and used texts to

students. In a recent biology class with 24 students,

14 bought used texts and 10 bought new texts, with

the class as a whole paying $2370. Of the 6 premed

students in class, 2 bought used texts, and 4 bought

new texts, with the group paying a total of $660.

How much does a used text cost? How much does a

new text cost?

CB⫺ 4A⫺2CD

D ⫽ £

20⫺6

⫺1 ⫺30

15⫺4

§C ⫽ c

4 ⫺8 ⫺3

⫺10 1

B ⫽ £

6 ⫺2

§A ⫽ c

4 ⫺1

0 ⫺5

⫺0.6DC ⫹

D ⫽ £

5 2.5 10

⫺2.5 0 ⫺5

10 2.5 10

§C ⫽ £

⫺0.2 0 0.2

0.4 0.8 0

0.1 ⫺0.2 ⫺0.1

MID-CHAPTER CHECK

cob19545_ch07_651-663.qxd 11/29/10 10:02 AM Page 662

7–27 Section 7.3 Solving Linear Systems Using Matrix Equations 663

College Algebra G&M—

10. Table A shown gives the number and type of extended warranties sold to individual car owners and to business

ﬂeets. Table B shows the promotions offered to those making the purchase. Write the entries of each table in

matrix form and compute the product matrix , then state what each entry of the product matrix represents.P ⫽ AB

Table A

Extended 80,000 100,000 120,000

Warranties mi mi mi

Individuals 30 25 10

Businesses 20 12 5

Table B

Free AAA

Promotions Rebate Membership

80,000 mi $50 1 yr

100,000 mi $75 2 yr

120,000 mi $100 3 yr

REINFORCING BASIC CONCEPTS

More on Matrix Multiplication

To help understand and master the concept of matrix multiplication, it helps to take a closer

look at the entries of the product matrix. Recall for the product , the entry p

in the

product matrix is the result of multiplying the 1st row of A with the 1st column of B, the entry

is the result of multiplying 1st row of A, with the 2nd column of B, and so on.

Exercise 1: The product of the 3rd row of A with the 2nd column of B, gives what entry in P?

Exercise 2: The entry p

is the result of what product? The entry p

is the result of what product?

Exercise 3: If p

is the last entry of the product matrix, what are the possible sizes of A and B?

Exercise 4: Of the eight matrices shown here, only two produce the product matrix P shown. Use the ideas

highlighted above to determine which two.

H ⫽ 31 34G ⫽ £

§F ⫽ £

§E ⫽ c

3 ⫺1

D ⫽ £

⫺10

§C ⫽ £

20⫺6

⫺1 ⫺30

15⫺4

§B ⫽ c

dA ⫽ £

3 ⫺1

AB ⫽ P

P ⫽ £

6 ⫺2

7.3 Solving Linear Systems Using Matrix Equations

While using matrices and row operations offers a degree of efﬁciency in solving sys-

tems, we are still required to solve for each variable individually. Using matrix multipli-

cation we can actually rewrite a given system as a single matrix equation, in which the

solutions are computed simultaneously. As with other kinds of equations, the use of

identities and inverses are involved, which we now develop in the context of matrices.

A. Multiplication and Identity Matrices

From the properties of real numbers, 1 is the identity for multiplication since

A similar identity exists for matrix multiplication. Consider the

matrix While matrix multiplication is not generally commutative, A ⫽ c

⫺23

2 ⫻ 2n

1 ⫽ 1

n ⫽ n.

LEARNING OBJECTIVES

In Section 7.3 you will see

how we can:

A. Recognize the identity

matrix for multiplication

B. Find the inverse of a

square matrix

C. Solve systems using

matrix equations

D. Use determinants to find

whether a matrix is

invertible

cob19545_ch07_651-663.qxd 10/21/10 9:48 PM Page 663

College Algebra G&M—

if we can ﬁnd a matrix B where then B is a prime candidate for the

identity matrix, which is denoted I. For the products AB and BA to be possible and have

the same order as A, we note B must also be a matrix. Using the arbitrary matrix

we have the following.

EXAMPLE 1A

䊳

Solving AB  A to Find the Identity Matrix

For use matrix multiplication, the equality of

matrices, and systems of equations to ﬁnd the value of a, b, c, and d.

Solution

䊳

The product on the left gives

Since corresponding entries must be equal (shown by matching colors), we can ﬁnd

a, b, c, and d by solving the systems and . For

the ﬁrst system, shows and Using for the second

shows and It appears is a candidate for the identity matrix.

Before we name B as the identity matrix, we must show that

EXAMPLE 1B

䊳

Verifying AB  BA  A

Given and determine if and

Solution

䊳

✓✓

Since B is the identity matrix I.

Now try Exercises 7 through 10

䊳

By replacing the entries of with those of the general matrix

we can show that is the identity for all matrices. In

considering the identity for larger matrices, we ﬁnd that only square matrices have in-

verses, since is the primary requirement (the multiplication must be possible

in both directions). This is commonly referred to as multiplication from the right and

multiplication from the left. Using the same procedure as before we can show

is the identity for matrices (denoted I

). The identity matrix

consists of 1’s down the main diagonal and 0’s for all other entries. Also, the iden-

tity I

for a square matrix is unique.

n ⫻ n3 ⫻ 3£

100

010

001

AI ⫽ IA

2 ⫻ 2I ⫽ c

A ⫽ c

⫺2 ⫺3

AB ⫽ A ⫽ BA,

⫽ c

⫺23

d⫽ A ⫽ c

⫺23

d⫽ A

⫽ c

1112⫹ 01⫺22 1142⫹ 0132

0112⫹ 11⫺22 0142⫹ 1132

d ⫽ c

1112⫹ 4102

1102⫹ 4112

⫺2112⫹ 3102 ⫺2102⫹ 3112

BA ⫽ c

d c

⫺23

d AB ⫽ c

⫺23

d c

BA ⫽ A.AB ⫽ AB ⫽ c

d,A ⫽ c

⫺23

AB ⫽ BA ⫽ A.

dd ⫽ 1.b ⫽ 0

2R1 ⫹ R2c ⫽ 0.a ⫽ 12R1 ⫹ R2

b ⫹ 4d ⫽ 4

⫺2b ⫹ 3d ⫽ 3

a ⫹ 4c ⫽ 1

⫺2a ⫹ 3c ⫽ ⫺2

a  4c b ⫹ 4d

⫺2a ⫹ 3c ⫺2b ⫹ 3d

d⫽ c

1 4

⫺2 3

⫺23

d c

d⫽ c

⫺23

B ⫽ c

2 ⫻ 2

AB ⫽ BA ⫽ A,

664 CHAPTER 7 Matrices and Matrix Applications 7–28

cob19545_ch07_664-679.qxd 12/1/10 6:19 PM Page 664

As in Section 7.2, a graphing calculator can

be used to investigate operations on matrices

and matrix properties. For the matrix

and a

calculator will conﬁrm that

Carefully enter A into your calculator as

matrix A, and I

as matrix B. Figure 7.14

shows and . See Exercises 11

through 14.

B. The Inverse of a Matrix

Again from the properties of real numbers, we know the multiplicative inverse for a is

since the products and yield the identity 1. To show

that a similar inverse exists for matrices, consider the square matrix and

an arbitrary matrix If we can ﬁnd a matrix B, where then B

is a prime candidate for the inverse matrix of A, which is denoted In an attempt to

ﬁnd such a matrix B, we proceed as in Examples 1A and 1B.

EXAMPLE 2A

䊳

Solving AB  I to ﬁnd A

1

For use matrix multiplication, the equality of matrices,

and systems of equations to ﬁnd the entries of B.

Solution

䊳

The product on the left gives Since corresponding

entries must be equal (shown by matching colors), we ﬁnd the values of a, b, c, and

d by solving the systems and Using

for the ﬁrst system shows and while for the second

system shows and Matrix is the

prime candidate for

It may have occurred to you that the systems used in Example 2A have exactly the

same coefﬁcients. In matrix form, these systems are and . In-

stead of solving these two systems separately, we can solve them simultaneously by simply

using the coefﬁcient matrix augmented with the identity matrix: .

Row operations are then used to transform the given matrix into the identity:

, in a sense “preserving” the inverse operations needed. The inverse

matrix is then .c

10ef

01gh

6510

2201

650

221

651

220

1

B  c

d c

1 2.5

13

dd  3.b 2.5

3R2  R1c 1,a  1

3R2  R1e

6b  5d  0

2b  2d  1

6a  5c  1

2a  2c  0

6a  5c 6b  5d

2a  2c 2b  2d

d c

1 0

0 1

d c

d c

1

AB  BA  I,B  c

A  c

1



1a  02,

BA  AAB  A

 A  I

 £

100

010

001

§,A  £

25 1

4 11

032

3  3

7–29 Section 7.3 Solving Linear Systems Using Matrix Equations 665

College Algebra G&M—

A. You’ve just seen how

we can recognize the identity

matrix for multiplication

Figure 7.14

cob19545_ch07_664-679.qxd 10/21/10 9:50 PM Page 665

666 CHAPTER 7 Matrices and Matrix Applications 7–30

College Algebra G&M—

EXAMPLE 2B

䊳

Finding a 2  2 Inverse Using the Augmented Matrix

Use the augmented matrix method to ﬁnd for . Then use a

calculator to verify .

Solution

䊳

First we augment the given matrix with the identity: . Then we use

row operations to transform A into the identity matrix.

As in Example 2A, a prime candidate for . Using a calculator

with and conﬁrms that (see ﬁgure).

Now try Exercises 15 through 22

䊳

These observations guide us to the following deﬁnition of an inverse matrix.

The Inverse of a Matrix

Given an matrix A, if there exists an matrix

such that , then is the inverse of matrix A.

We will soon discover that while only square matrices have inverses, not every

square matrix has an inverse. If an inverse exists, the matrix is said to be invertible. For

matrices that are invertible, a simple formula exists for computing the inverse.

The formula is derived in the Strengthening Core Skills feature at the end of Chapter 7.

The Inverse of a 2  2 Matrix

If then provided

To “test” the formula, again consider the matrix where

and

✓

See Exercises 63 through 66 for more practice with this formula.



2 5

26

d c

1 2.5

13

1



162122 152122

2 5

26

d  2:c  2,

a  6, b  5,A  c

ad  bc  0A

1



ad  bc

d b

ca

dA  c

2  2

1

 A

1

A  I

1

n  nn  n

1

A  AA

1

 IA

1

 3B4A  3A4

1

is c

1 2.5

13

1R2 S R2

10 1 2.5

0113

S R1

60615

0 11 3

60615

0 11 3

d5R2  R1 S R1c

6510

0 113

6510

0 113

d3R2  R1 S R2c

6510

2201

6510

2201

1

A  AA

1

 I

A  c

1

cob19545_ch07_664-679.qxd 10/21/10 9:50 PM Page 666

Almost without exception, real-world applications involve much larger matrices,

with entries that are not integer-valued. Although the augmented matrix method from

Example 2 can be extended to ﬁnd the inverse of larger matrices, the process becomes

very tedious and too time consuming to be useful. The process for a 3  3 matrix is

discussed in the Strengthening Core Skills feature at the end of Chapter 7. But for

practical reasons, we will rely on a calculator to produce these larger inverse matrices.

This is done by (1) carefully entering a square matrix A into the calculator, (2) return-

ing to the home screen, and (3) calling up matrix A and pressing the key and

to ﬁnd . In the context of matrices, calculators are programmed to compute an in-

verse matrix using this key, rather than to somehow ﬁnd a reciprocal. See Exercises

23 through 26.

C. Solving Systems Using Matrix Equations

One reason matrix multiplication has its deﬁnition is to assist in writ-

ing a linear system of equations as a single matrix equation. The equation consists of

the matrix of constants B on the right, and a product of the coefﬁcient matrix A with the

matrix of variables X on the left: For the matrix

equation is Note that computing the product on the left

will yield the original system.

Once written as a matrix equation, the system can be solved using an inverse ma-

trix and the following sequence. If A represents the matrix of coefﬁcients, X the matrix

of variables, B the matrix of constants, and I the appropriate identity, the sequence is

matrix equation

multiply from the left by the inverse of

associative property

1



Lines 1 through 5 illustrate the steps that make the method work. In actual prac-

tice, after carefully entering the matrices, only step 5 is used when solving matrix

equations using technology. Once matrix A is entered, the calculator will automatically

ﬁnd and use as we enter

EXAMPLE 3

䊳

Using Technology to Solve a Matrix Equation

Use a calculator and a matrix equation to solve the system

Solution

䊳

As before, the matrix equation is £

141

253

812

§ £

§ £

§.

•

x  4y  z  10

2x  5y  3z  7.

8x  y  2z  11

1

B.A

1

152

X  A

1

142

IX  A

1

132

1

A2X  A

1

122

1

1AX2 A

1

112

AX  B

141

253

812

§ £

§ £

§.

•

x  4y  z  10

2x  5y  3z  7

8x  y  2z  11

,AX  B.

row  column

1

ENTER

-1

7–31 Section 7.3 Solving Linear Systems Using Matrix Equations 667

College Algebra G&M—

B. You’ve just seen how

we can ﬁnd the inverse of a

square matrix

cob19545_ch07_664-679.qxd 10/21/10 9:50 PM Page 667

Carefully enter (and double-check) the matrix of coefﬁcients as matrix A in your

calculator, and the matrix of constants as matrix B (see Figure 7.15). The product

shows the solution is (see Figure 7.16). Verify by

substitution.

Now try Exercises 27 through 44

䊳

The matrix equation method does have a few shortcomings. Consider the system

whose corresponding matrix equation is After entering

the matrix of coefﬁcients A and matrix of constants B, attempting to compute

results in the error message shown in Figure 7.17. The calculator is unable to return a

solution due to something called a “singular matrix.” To investigate further, we attempt

to ﬁnd for using the formula for a matrix. With

and we have

Since division by zero is undeﬁned, we conclude that matrix A has no inverse. A

matrix having no inverse is said to be singular or noninvertible. Solving systems

using matrix equations is only possible when the matrix of coefﬁcients is nonsingular.

D. Determinants and Singular Matrices

As a practical matter, it becomes important to know ahead of time whether a particular

matrix has an inverse. To help with this, we introduce one additional operation on a

square matrix, that of calculating its determinant. For a matrix the determinant

is the entry itself. For a matrix the determinant of A, written

as det(A) or denoted with vertical bars as is computed as a difference of diagonal

products beginning with the upper-left entry:

2nd diagonal

product

1st diagonal

product

The Determinant of a 2  2 Matrix

Given any matrix ,

det1A2

冟

 a

 a

A  c

d2  2

det1A2 `

` a

 a

冟

A  c

d,2  2

1  1



510

d A

1



ad  bc

d b

ca

d

142152 1102122

510

d  5,b 10, c 2,

a  4,2  2c

4 10

25

1

4 10

25

d c

d c

8

Figure 7.15 Figure 7.16

x  2, y  3, z  4A

1

668 CHAPTER 7 Matrices and Matrix Applications 7–32

College Algebra G&M—

C. You’ve just seen how

we can solve systems using

matrix equations

Figure 7.17

cob19545_ch07_664-679.qxd 10/21/10 9:50 PM Page 668

EXAMPLE 4

䊳

Calculating Determinants

Compute the determinant of each matrix.

a. b. c.

Solution

䊳

b. Determinants are only deﬁned for square matrices (see ﬁgure).

Now try Exercises 45 through 48

䊳

Notice from Example 4(c), the determinant of is zero, and this is the

same matrix we earlier found had no inverse. This observation can be extended to

larger matrices and offers the connection we seek between a given matrix, its inverse,

and matrix equations.

Singular Matrices

If A is a square matrix and the inverse matrix does not exist

and A is said to be singular or noninvertible.

In summary, inverses exist only for square matrices, but not every square matrix

has an inverse. If the determinant of a square matrix is zero, an inverse does not exist

and the method of matrix equations cannot be used to solve the system.

To use the determinant test for a system, we need to compute a deter-

minant. At ﬁrst glance, our experience with determinants appears to be of little

help. However, every entry in a matrix is associated with a smaller matrix,

formed by deleting the row and column of that entry and using the entries that remain.

These ’s are called the associated minor matrices or simply the minors. Using

a general matrix of coefﬁcients, we’ll identify the minors associated with the entries in

the ﬁrst row.

Entry: a

associated minor associated minor associated minor

To illustrate, consider the system shown, and (1) form the matrix of coefﬁcients,

(2) identify the minor matrices associated with the entries in the first row, and

(3) compute the determinant of each minor.

§£

2  2

2  23  3

2  2

3  33  3

det1A2 0,

4 10

25

det1D2 `

4 10

25

` 142152 1221102 20  20  0

det1B2 `

1 6

` 132162 11212220

D  c

4 10

25

dC  c

521

1 34

dB  c

1 6

7–33 Section 7.3 Solving Linear Systems Using Matrix Equations 669

College Algebra G&M—

WORTHY OF NOTE

For the determinant of a general

matrix using cofactors, see

Appendix IV.

n  n

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