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D. Determinants and Singular Matrices

D. Determinants and Singular Matrices

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Section 7.3 Solving Linear Systems Using Matrix Equations



EXAMPLE 4







Calculating Determinants

Compute the determinant of each matrix.

3 2

5

2 1

a. B ϭ c

b. C ϭ c

d

d

1 Ϫ6

Ϫ1 Ϫ3 4



Solution







669



c. D ϭ c



4

Ϫ2



Ϫ10

d

5



a. det1B2 ϭ `



3 2

` ϭ 1321Ϫ62 Ϫ 112122 ϭ Ϫ20

1 Ϫ6

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

4 Ϫ10

c. det1D2 ϭ `

` ϭ 142 152 Ϫ 1Ϫ22 1Ϫ102 ϭ 20 Ϫ 20 ϭ 0

Ϫ2

5



Now try Exercises 45 through 48







4 Ϫ10

d is zero, and this is the

Ϫ2

5

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.

Notice from Example 4(c), the determinant of c



Singular Matrices

If A is a square matrix and det1A2 ϭ 0, the inverse matrix does not exist

and A is said to be singular or noninvertible.



WORTHY OF NOTE

For the determinant of a general

n ϫ n matrix using cofactors, see

Appendix IV.



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 3 ϫ 3 system, we need to compute a 3 ϫ 3 determinant. At first glance, our experience with 2 ϫ 2 determinants appears to be of little

help. However, every entry in a 3 ϫ 3 matrix is associated with a smaller 2 ϫ 2 matrix,

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

These 2 ϫ 2’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.

a11 a12

£ a21 a22

a31 a32



a13

a23 §

a33



Entry: a11

associated minor

a22 a23

c

d

a32 a33



a11

£ a21

a31



a12 a13

a22 a23 §

a32 a33



Entry: a12

associated minor

a21 a23

c

d

a31 a33



a11

£ a21

a31



a12 a13

a22 a23 §

a32 a33



Entry: a13

associated minor

a21 a22

d

c

a31 a32



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.



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CHAPTER 7 Matrices and Matrix Applications



2x ϩ 3y Ϫ z ϭ 1

• x Ϫ 4y ϩ 2z ϭ Ϫ3

ϭ Ϫ1

3x ϩ y

2

122 £ 1

3



3 Ϫ1

Ϫ4 2 §

1

0



2

(1) Matrix of coefficients £ 1

3



3 Ϫ1

Ϫ4



1

0



2

£1

3



3

Ϫ4

1



Ϫ1



0



2

£1

3



Ϫ1



0



3

Ϫ4

1



Entry a11: 2

associated minor

Ϫ4 2

c

d

1 0



Entry a12: 3

associated minor

1 2

c

d

3 0



Entry a13: Ϫ1

associated minor

1 Ϫ4

c

d

3 1



(3) Determinant

of minor



Determinant of

minor



Determinant of

minor



1Ϫ42 102 Ϫ 112122 ϭ Ϫ2



112 102 Ϫ 132122 ϭ Ϫ6 112 112 Ϫ 1321Ϫ42 ϭ 13



For computing a 3 ϫ 3 determinant, we illustrate a technique called expansion

by minors.

The Determinant of a 3 ؋ 3 Matrix — Expansion by Minors

For the matrix M shown, det(M) is the unique number computed

matrix M

as follows:

a11 a12 a13

1. Select any row or column and form the product of each

£ a21 a22 a23 §

entry with its minor matrix. The illustration here uses the

a31 a32 a33

entries in row 1:

det1M2 ϭ ϩa11 `



a22

a32



a23

a21 a23

a21

` Ϫ a12 `

` ϩ a13 `

a33

a31 a33

a31



a22

`

a32



2. The signs used between terms of the expansion depends

on the row or column chosen, according to the sign chart

shown.



Sign Chart

ϩ

£ Ϫ

ϩ



Ϫ

ϩ

Ϫ



ϩ

Ϫ §

ϩ



The determinant of a matrix is unique and any row or column can be used. For this

reason, it’s helpful to select the row or column having the most zero, positive, and/or

smaller entries.

EXAMPLE 5







Calculating a 3 ؋ 3 Determinant

2

1

Compute the determinant of M ϭ £ 1 Ϫ1

Ϫ2 1



Solution







Ϫ3

0 §.

4



Since the second row has the “smallest” entries as well as a zero entry, we compute

the determinant using this row. According to the sign chart, the signs of the terms

will be negative–positive–negative, giving

1 Ϫ3

2 Ϫ3

2 1

` ϩ 1Ϫ12 `

` Ϫ 102 `

`

1 4

Ϫ2 4

Ϫ2 1

ϭ Ϫ114 ϩ 32 ϩ 1Ϫ1218 Ϫ 62 Ϫ 10212 ϩ 22

ϭ

Ϫ7

ϩ

(Ϫ2) Ϫ

0

ϭ Ϫ9

The value of det1M2 is Ϫ9.



det1M2 ϭ Ϫ112 `



Now try Exercises 49 through 52







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Section 7.3 Solving Linear Systems Using Matrix Equations



Try computing the determinant of M two more times, using a different row or column each time. Since the determinant is unique, you should obtain the same result.

There are actually other alternatives for computing a 3 ϫ 3 determinant. The first

is called determinants by column rotation, and takes advantage of patterns generated

from the expansion of minors. This method is applied to the matrix shown, which uses

alphabetical entries for simplicity.

a

det £ d

g



b

e

h



c ϭ a1ei Ϫ fh2 Ϫ b1di Ϫ fg2 ϩ c1dh Ϫ eg2

f § ϭ aei Ϫ afh Ϫ bdi ϩ bfg ϩ cdh Ϫ ceg

i

ϭ aei ϩ bfg ϩ cdh Ϫ afh Ϫ bdi Ϫ ceg



expansion using R1

distribute

rewrite result



Although history is unsure of who should be credited, notice that if you repeat the

first two columns to the right of the given matrix (“rotation of columns”), identical

products are obtained using the six diagonals formed—three in the downward direction using addition, three in the upward direction using subtraction.

a

£d

g



b

e

h



gec

c



i

aei



hfa idb

a b

d e

g h

bfg cdh



Adding the products in blue (regardless of sign) and subtracting the products in

red (regardless of sign) gives the determinant. This method is more efficient than

expansion by minors, but can only be used for 3 ϫ 3 matrices!

EXAMPLE 6







Calculating det(A) Using Column Rotation

1

Use the column rotation method to find the determinant of A ϭ £ Ϫ2

Ϫ3



Solution







5

3

Ϫ8 0 § .

Ϫ11 1



Rotate columns 1 and 2 to the right, and compute the diagonal products.

1

£ Ϫ2

Ϫ3



72

0

Ϫ10

1

5

5

3

Ϫ8 0 § Ϫ2 Ϫ8

Ϫ3 Ϫ11

Ϫ11 1

Ϫ8

0

66



Adding the products in blue (regardless of sign) and subtracting the products in

red (regardless of sign) shows det1A2 ϭ Ϫ4:

Ϫ8 ϩ 0 ϩ 66 Ϫ 72 Ϫ 0 Ϫ 1Ϫ102 ϭ Ϫ4.



Now try Exercises 53 through 56







The final method is presented in the Extending the Concept feature of the Exercise

Set, and shows that if certain conditions are met, the determinant of a matrix can be

found using its triangularized form.

As with the operations studied in Section 7.2, the process of computing a determinant becomes very cumbersome for larger matrices, or those with rational or radical

entries. Most graphing calculators are programmed to handle these computations easily.

x

After accessing the matrix menu ( 2nd

), calculating a determinant is the first

option under the MATH submenu (Figure 7.18). The calculator results for det([A]) and

det([B]) as defined are shown in Figures 7.19 and 7.20. See Exercises 57 and 58.

-1



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CHAPTER 7 Matrices and Matrix Applications



Figure 7.18



EXAMPLE 7







Figure 7.19



Figure 7.20



Solving a System after Verifying A is Invertible

Given the system shown here, (1) form the matrix equation AX ϭ B; (2) compute

the determinant of the coefficient matrix and determine if you can proceed; and

(3) if so, solve the system using a matrix equation.

2x ϩ 1y Ϫ 3z ϭ 11

• 1x Ϫ 1y

ϭ1

Ϫ2x ϩ 1y ϩ 4z ϭ Ϫ8



Solution







1. Form the matrix equation AX ϭ B:

2

1

£ 1 Ϫ1

Ϫ2 1



Ϫ3 x

11

0 § £y§ ϭ £ 1 §

4

z

Ϫ8



2. Enter the matrices A and B into the calculator. Since det(A) is nonzero (from

Example 5 and Figure 7.21), we can proceed.

3. Enter AϪ1B on the home screen and press

(Figure 7.22).

ENTER



Figure 7.21



Figure 7.22



The solution is the ordered triple 13, 2, Ϫ12 .

Now try Exercises 59 through 62







We close this section with an application involving a 4 ϫ 4 system. There is a

large variety of additional applications in the Exercise Set.

EXAMPLE 8







Solving an Application Using Technology and Matrix Equations

A local theater sells four sizes of soft drinks: 32 oz @ $2.25; 24 oz @ $1.90; 16 oz

@ $1.50; and 12 oz @ $1.20/each. As part of a “free guest pass” promotion, the

manager asks employees to try and determine the number of each size sold, given

the following information: (1) the total revenue from soft drinks was $719.80;

(2) there were 9096 oz of soft drink sold; (3) there was a total of 394 soft drinks

sold; and (4) the number of 24-oz and 12-oz drinks sold was 12 more than the

number of 32-oz and 16-oz drinks sold. Write a system of equations that models

this information, then solve the system using a matrix equation.



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Section 7.3 Solving Linear Systems Using Matrix Equations



Solution







673



If we let x, l, m, and s represent the number of 32-oz, 24-oz, 16-oz, and 12-oz soft

drinks sold, the following system is produced:

2.25x ϩ 1.90l ϩ 1.50m ϩ 1.20s ϭ 719.8

revenue:

32x ϩ 24l ϩ 16m ϩ 12s ϭ 9096

ounces sold:

μ

x ϩ l ϩ m ϩ s ϭ 394

quantity sold:

l ϩ s ϭ x ϩ m ϩ 12

amounts sold:

When written as a matrix equation the system becomes:

2.25

32



1

Ϫ1



1.9

24

1

1



1.5 1.2

x

719.8

16 12

l

9096

¥ ≥ ¥ ϭ ≥

¥

1

1

m

394

Ϫ1 1

s

12



To solve, carefully enter the matrix of coefficients as matrix A (see Figure 7.23), and

the matrix of constants as matrix B, then compute AϪ1B ϭ X [since det1A2 0 4.

This gives a solution of 1x, l, m, s2 ϭ 1112, 151, 79, 522 1Figure 7.242.

Figure 7.23



D. You’ve just seen how

we can use determinants to

find whether a matrix is

invertible



Figure 7.24



Now try Exercises 67 through 78







7.3 EXERCISES





CONCEPTS AND VOCABULARY



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



1. The n ϫ n identity matrix In consists of 1’s down

the

and

for all other entries.



2. The product of a square matrix A and its inverse

AϪ1 yields the

matrix.



3. Given square matrices A and B of like size, B is the

inverse of A if

ϭ

ϭ

. Notationally we

write B ϭ

.



4. If the determinant of a matrix is zero, the matrix is

said to be

or

, meaning no

inverse exists.



5. Explain why inverses exist only for square

matrices, then discuss why some square matrices

do not have an inverse. Illustrate each point with an

example.



6. What is the connection between the determinant of

a 2 ϫ 2 matrix and the formula for finding its

inverse? Use the connection to create a 2 ϫ 2

matrix that is invertible, and another that is not.



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CHAPTER 7 Matrices and Matrix Applications



DEVELOPING YOUR SKILLS



Use matrix multiplication, equality of matrices, and the

a b

1 0

arbitrary matrix given to show that c

d ‫ ؍‬c

d.

c d

0 1



2

7. A ϭ c

Ϫ3



5

a b

2

5

dc

d ϭ c

d

Ϫ7 c d

Ϫ3 Ϫ7



8. A ϭ c



9

Ϫ5



Ϫ7 a

dc

4

c



b

9

d ϭ c

d

Ϫ5



Ϫ7

d

4



9. A ϭ c



0.4

0.3



0.6 a

dc

0.2 c



b

0.4

d ϭ c

d

0.3



0.6

d

0.2



1



1

4

1d

8



10. A ϭ c 12

3



For I2 ‫ ؍‬c



1

0



c



a

c



1

b

d ϭ c 21

d

3



1

0

d , I3 ‫ ؍‬£ 0

1

0



0

1

0



1

4

1d

8



Ϫ3

Ϫ4



8

d

10



0

0 § , and

1



Ϫ4 1

13. £ 9

5

0 Ϫ2



12. c

6



1



0.5

Ϫ0.7



9

2

14. ≥

4

0



Ϫ0.2

d

0.3



1

3 Ϫ1

0 Ϫ5 3

¥

6

1

0

Ϫ2 4

1



Find the inverse of each 2 ؋ 2 matrix using matrix

multiplication, equality of matrices, and a system of

equations.



15. c



5

2



Ϫ4

d

2



16. c



1

0



Ϫ5

d

Ϫ4



Find the inverse of each matrix by augmenting of the

the identity matrix and using row operations.



17. c



1

4



Ϫ3

d

Ϫ10



18. c



4

0



Ϫ5

d

2



1



5

8

1d

2



Bϭ c4

0



Ϫ2 0.4

d

1 0.8



Demonstrate that B ‫ ؍‬A؊1, by showing AB ‫ ؍‬BA ‫ ؍‬I.

Do not use a calculator.



19. A ϭ c



1

Ϫ2



5

d

Ϫ9



20. A ϭ c



Ϫ2 Ϫ6

d

4

11



Bϭ c



Ϫ9

2



Ϫ5

d

1



Bϭ c



5.5 3

d

Ϫ2 Ϫ1



22. A ϭ c



Ϫ2 5

d

3 Ϫ4

4



B ϭ c 37

7



5

7

2d

7



Use a calculator to find A؊1 ‫ ؍‬B, then confirm the

inverse by showing AB ‫ ؍‬BA ‫ ؍‬I.



Ϫ2 3 1

23. A ϭ £ 5 2 4 §

2 0 Ϫ1

0.5

24. A ϭ £ 0

1



1 0 0 0

0 1 0 0

I4 ‫≥ ؍‬

¥ , show AI ‫ ؍‬IA ‫ ؍‬A for the

0 0 1 0

0 0 0 1

matrices of like size. Use a calculator for Exercise 14.



11. c



21. A ϭ c



0.2

0.3

0.4



0.1

0.6 §

Ϫ0.3



Ϫ7 5 Ϫ3

25. A ϭ Ê 1

9

0 Đ

2 2 5

12

1

0

26. A



12 12

0



6

4

12

12



3

0

8 12

Ơ

0

0

0 12



Write each system in the form of a matrix equation. Do

not solve.



27. e



2x Ϫ 3y ϭ 9

Ϫ5x ϩ 7y ϭ 8



28. e



0.5x Ϫ 0.6y ϭ 0.6

Ϫ0.7x ϩ 0.4y ϭ Ϫ0.375



x ϩ 2y Ϫ z ϭ 1

29. • x ϩ z ϭ 3

2x Ϫ y ϩ z ϭ 3

2x Ϫ 3y Ϫ 2z ϭ 4

30. • 14x Ϫ 25y ϩ 34z ϭ Ϫ1

3

Ϫ2x ϩ 1.3y Ϫ 3z ϭ 5

Ϫ2w ϩ x Ϫ 4y ϩ 5z ϭ Ϫ3

2w Ϫ 5x ϩ y Ϫ 3z ϭ 4

31. μ

Ϫ3w ϩ x ϩ 6y ϩ z ϭ 1

w ϩ 4x Ϫ 5y ϩ z ϭ Ϫ9

1.5w ϩ 2.1x Ϫ 0.4y ϩ z ϭ 1

0.2w Ϫ 2.6x ϩ y ϭ 5.8

32. μ

3.2x ϩ z ϭ 2.7

1.6w ϩ 4x Ϫ 5y ϩ 2.6z ϭ Ϫ1.8



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Write each system as a matrix equation and solve (if

possible) using inverse matrices and your calculator. If

the coefficient matrix is singular, write no solution.



33. e



0.05x Ϫ 3.2y ϭ Ϫ15.8

0.02x ϩ 2.4y ϭ 12.08



34. e



0.3x ϩ 1.1y ϭ 3.5

Ϫ0.5x Ϫ 2.9y ϭ Ϫ10.1



35.



12a ϩ 13b ϭ 216

16a ϩ

b ϭ 412



37. e



Ϫ5

3

3

2 a ϩ 5 b ϭ Ϫ10

3

7

5

16 a Ϫ 2 b ϭ Ϫ16



38. e



3 12a ϩ 213b ϭ 12

512a Ϫ 313b ϭ 1



1

52. D ϭ £ 2.5

3



4x Ϫ 5y Ϫ 6z ϭ 33

Ϫ 35y ϩ 54z ϭ 9

Ϫ0.5x ϩ 2.4y Ϫ 4z ϭ Ϫ32



Ϫ2w ϩ 3x Ϫ 4y ϩ 5z ϭ Ϫ3

0.2w Ϫ 2.6x ϩ

y Ϫ 0.4z ϭ 2.4

43. μ

Ϫ3w ϩ 3.2x ϩ 2.8y ϩ z ϭ 6.1

1.6w ϩ 4x Ϫ 5y ϩ 2.6z ϭ Ϫ9.8

2w Ϫ 5x ϩ 3y Ϫ 4z ϭ 7

1.6w ϩ 4.2y Ϫ 1.8z ϭ 5.4

44. μ

3w ϩ 6.7x Ϫ 9y ϩ 4z ϭ Ϫ8.5

0.7x Ϫ 0.9z ϭ 0.9

Compute the determinant of each matrix and state

whether an inverse matrix exists. Do not use a

calculator.



1.2

0.3



Ϫ0.8

d

Ϫ0.2



Ϫ0.8

Ϫ2 §

Ϫ2.5



2

53. £ 4

1



Ϫ3 1

Ϫ1 5 §

0 Ϫ2



1

55. £ 3

4



Ϫ1 2

Ϫ2 4 §

3 1



54.



Ϫ3

£ 1

3



56.



5 6 2

Ê 2 1 2 Đ

3 4 1



-1



18x



47. c



4

2 §

Ϫ2



2 4

Ϫ2 0 §

1 5



Use a calculator to compute the determinant of each

matrix. If the determinant is zero, write singular matrix.

If the determinant is nonzero, find A؊1 and store the

result as matrix B ( STO 2nd x 2: [B] ). Then

verify the inverse by showing AB ‫ ؍‬BA ‫ ؍‬I.



x Ϫ 2y ϩ 2z ϭ 9

41. • 2x Ϫ 1.5y ϩ 1.8z ϭ 12

Ϫ2

1

3

3 x ϩ

2y Ϫ

5 z ϭ Ϫ4



Ϫ7

d

Ϫ5



2

5

0



1



0



Compute the determinant of each matrix using the

column rotation method.



1.7x ϩ 2.3y Ϫ 2z ϭ 41.5

40. • 1.4x Ϫ 0.9y ϩ 1.6z ϭ Ϫ10

Ϫ0.8x ϩ 1.8y Ϫ 0.5z ϭ 16.5



4

3



0 Ϫ2

Ϫ2 2

Ϫ1 Ϫ1 § 50. B ϭ £ 0 Ϫ1

1 Ϫ4

4 Ϫ4



Ϫ2

3

51. C ϭ £ 0

6

1 Ϫ1.5



0.2x Ϫ 1.6y ϩ 2z ϭ Ϫ1.9

39. • Ϫ0.4x Ϫ y ϩ 0.6z ϭ Ϫ1

0.8x ϩ 3.2y Ϫ 0.4z ϭ 0.2



45. c



Compute the determinant of each matrix without using

a calculator. If the determinant is zero, write singular

matrix.



1

49. A ϭ £ 0

2



Ϫ1

u ϩ 14v ϭ 1

e 16

2

2 u Ϫ 3 v ϭ Ϫ2



36. e



42.



675



Section 7.3 Solving Linear Systems Using Matrix Equations



46. c



0.6

0.4



0.3

d

0.5



48. c



Ϫ2

Ϫ3



6

d

9



1

2

57. A ϭ ≥

8

0



0

5

15

8



3

0

6

Ϫ4



1

2

0

1

58. M ϭ ≥

Ϫ1 0

2 Ϫ1



ENTER



Ϫ4

1

¥

Ϫ5

1

1

Ϫ3

2

1



1

2

¥

Ϫ3

4



For each system shown, form the matrix equation

AX ‫ ؍‬B; compute the determinant of the coefficient

matrix and determine if you can proceed; and if

possible, solve the system using the matrix equation.



x Ϫ 2y ϩ 2z ϭ 7

2x Ϫ 3y Ϫ 2z ϭ 7

59. • 2x ϩ 2y Ϫ z ϭ 5 60. • x Ϫ y ϩ 2z ϭ Ϫ5

3x Ϫ y ϩ z ϭ 6

3x ϩ 2y Ϫ z ϭ 11

x Ϫ 3y ϩ 4z ϭ Ϫ1

5x Ϫ 2y ϩ z ϭ 1

61. • 4x Ϫ y ϩ 5z ϭ 7 62. • 3x Ϫ 4y ϩ 9z ϭ Ϫ2

3x ϩ 2y ϩ z ϭ Ϫ3

4x Ϫ 3y ϩ 5z ϭ 6



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CHAPTER 7 Matrices and Matrix Applications



WORKING WITH FORMULAS



The inverse of a 2 ؋ 2 matrix: A ‫ ؍‬c



a b

1

#c d

d S A ؊1 ‫؍‬

c d

ad ؊ bc ؊c



؊b

d

a



The inverse of a 2 ؋ 2 matrix can be found using the formula shown, as long as ad ؊ bc 0. Use the formula to find

inverses for the matrices here (if possible), then verify by showing A # A؊1 ‫ ؍‬A # A؊1 ‫ ؍‬I.



63. A ϭ c





3

2



Ϫ5

d

1



64. B ϭ c



2

3

d

Ϫ5 Ϫ4



65. C ϭ c



0.3

Ϫ0.6



Ϫ0.4

d

0.8



66. c



0.2

Ϫ0.4



0.3

d

Ϫ0.6



APPLICATIONS



Solve each application using a matrix equation.

Descriptive Translation



67. Convenience store sales: The local Moto-Mart

sells four different sizes of Slushies — behemoth,

60 oz @ $2.59; gargantuan, 48 oz @ $2.29;

mammoth, 36 oz @ $1.99; and jumbo, 24 oz

@ $1.59. As part of a promotion, the owner offers

free gas to any customer who can tell how many of

each size were sold last week, given the following

information: (1) The total revenue for the Slushies

was $402.29; (2) 7884 ounces were sold; (3) a total

of 191 Slushies were sold; and (4) the number of

behemoth Slushies sold was one more than the

number of jumbo. How many of each size were sold?

68. Cartoon characters: In America, four of the most

beloved cartoon characters are Foghorn Leghorn,

Elmer Fudd, Bugs Bunny, and Tweety Bird.

Suppose that Bugs Bunny is four times as tall as

Tweety Bird. Elmer Fudd is as tall as the combined

height of Bugs Bunny and Tweety Bird. Foghorn

Leghorn is 20 cm taller than the combined height of

Elmer Fudd and Tweety Bird. The combined height

of all four characters is 500 cm. How tall is each one?

69. Rolling Stones music: One of the most prolific

and popular rock-and-roll bands of all time is the

Rolling Stones. Four of their many great hits

include: Jumpin’ Jack Flash, Tumbling Dice, You

Can’t Always Get What You Want, and Wild

Horses. The total playing time of all four songs is

20.75 min. The combined playing time of Jumpin’

Jack Flash and Tumbling Dice equals that of You

Can’t Always Get What You Want. Wild Horses is 2

min longer than Jumpin’ Jack Flash, and You Can’t

Always Get What You Want is twice as long as

Tumbling Dice. Find the playing time of each song.

70. Mozart’s arias: Mozart wrote some of vocal

music’s most memorable arias in his operas,



including Tamino’s Aria, Papageno’s Aria, the

Champagne Aria, and the Catalogue Aria. The

total playing time of all four arias is 14.3 min.

Papageno’s Aria is 3 min shorter than the

Catalogue Aria. The Champagne Aria is 2.7 min

shorter than Tamino’s Aria. The combined time of

Tamino’s Aria and Papageno’s Aria is five times

that of the Champagne Aria. Find the playing time

of all four arias.

Manufacturing



71. Resource allocation: Time Pieces Inc.

manufactures four different types of grandfather

clocks. Each clock requires these four stages:

(1) assembly, (2) installing the clockworks,

(3) inspection and testing, and (4) packaging for

delivery. The time required for each stage is shown

in the table, for each of the four clock types. At the

end of a busy week, the owner determines that

personnel on the assembly line worked for 262 hr,

the installation crews for 160 hr, the testing

department for 29 hr, and the packaging

department for 68 hr. How many clocks of each

type were made?

Dept.



Clock A



Clock B



Clock C



Clock D



Assemble



2.2



2.5



2.75



3



Install



1.2



1.4



1.8



2



Test



0.2



0.25



0.3



0.5



Pack



0.5



0.55



0.75



1.0



72. Resource allocation: Figurines Inc. makes and

sells four sizes of metal figurines, mostly historical

figures and celebrities. Each figurine goes through

four stages of development: (1) casting, (2) trimming,

(3) polishing, and (4) painting. The time required

for each stage is shown in the table, for each of the

four sizes. At the end of a busy week, the manager

finds that the casting department put in 62 hr, and



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Section 7.3 Solving Linear Systems Using Matrix Equations



the trimming department worked for 93.5 hr, with

the polishing and painting departments logging

138 hr and 358 hr, respectively. How many

figurines of each type were made?

Dept.



Small



Medium



Large



X-Large



Casting



0.5



0.6



0.75



1



Trimming



0.8



0.9



1.1



1.5



Polishing



1.2



1.4



1.7



2



Painting



2.5



3.5



4.5



6



64°C

73. Thermal conductivity:

In lab experiments

designed to measure

the heat conductivity

p2

p1

70°C

80°C

of a square metal plate

of uniform density, the

p3

p4

edges are held at four

different (constant)

96°C

temperatures. The

mean-value principle from physics tells us that the

temperature at a given point pi on the plate is equal

to the average temperature of nearby points. Use

this information to form a system of four equations

in four variables, and determine the temperature at

interior points p1, p2, p3, and p4 on the plate shown.

(Hint: Use the temperature of the four points closest

to each.)



74. Thermal conductivity: Repeat Exercise 73 if

(a) the temperatures at the top and bottom of the

plate were increased by 10°, with the temperatures

at the left and right edges decreased by 10° (what

do you notice?); (b) the temperature at the top and

the temperature to the left were decreased by 10°,

with the temperatures at the bottom and right held

at their original temperature.

Curve Fitting



75. Quadratic fit: Use a matrix equation to find a

quadratic function of the form y ϭ ax2 ϩ bx ϩ c

such that 1Ϫ4, Ϫ52 , 10, Ϫ52 , and (2, 7) are on the

graph of the function.

76. Quadratic fit: Use a matrix equation to find a

quadratic function of the form y ϭ ax2 ϩ bx ϩ c

such that 1Ϫ4, Ϫ02 , (1, 5) and, 12, Ϫ62 are on the

graph of the function.

77. Cubic fit: Use a matrix equation to find a cubic

function of the form y ϭ ax3 ϩ bx2 ϩ cx ϩ d such

that 1Ϫ4, Ϫ62, 1Ϫ1, 02, 11, Ϫ162, and (3, 8) are on

the graph of the function.

78. Cubic fit: Use a matrix equation to find a cubic

function of the form y ϭ ax3 ϩ bx2 ϩ cx ϩ d such

that 1Ϫ2, 52, 10, 12, 12, Ϫ32, and (3, 25) are on the

graph of the function.



677



Investing



79. Wise investing: Morgan received an $800 gift

from her grandfather, and showing wisdom beyond

her years, decided to place the money in a

certificate of deposit (CD) and a money market

fund (MM). At the time, CDs were earning 3.5%

and MMs were earning 2.5%. At the end of 1 yr,

she cashed both in and received a total of $824.50.

How much was deposited in each?

80. Baseball cards: Gary has a passion for baseball,

which includes a collection of rare baseball

cards. His most prized cards feature Willie Mays

(1953 Topps) and Mickey Mantle (1959 Topps).

The Willie Mays card has appreciated 28% and the

Mickey Mantle card 25% since he purchased them,

and together they are now worth $17,100. If he

paid a total of $13,507.50 at auction for both cards,

what was the original price of each?



Willie Mays



(1953 Topps) $7187.50



81. Retirement planning: Using payroll deduction,

Jeanette was able to put aside $4800 per month last

year for her impending retirement. Last year, her

company retirement fund paid 4.2% and her mutual

funds returned 5.75%, but her stock fund actually

decreased 2.5% in value. If her net gain for the

year was $104.50 and $300 more was placed in

stocks than in mutual funds, how much was placed

in each investment vehicle?

82. Charitable giving: The

hyperbolic funnels seen

at many shopping malls

are primarily used by

nonprofit organizations to

raise funds for worthy

causes. A coin is

launched down a ramp into the funnel and

seemingly makes endless circuits before finally

disappearing down a “black hole” (the collection

bin). During one such collection, the bin was found

to hold $112.89, and 1450 coins consisting of

pennies, nickels, dimes, and quarters. How many

of each denominator were there, if the number of

quarters and dimes was equal to the number of

nickels, and the number of pennies was twice

the number of quarters.



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7–42



CHAPTER 7 Matrices and Matrix Applications



Nutrition



83. Animal diets: A zoo dietician needs to create a specialized diet that

regulates an animal’s intake of fat, carbohydrates, and protein during

a meal. The table given shows three different foods and the amount

of these nutrients (in grams) that each ounce of food provides. How

many ounces of each should the dietician recommend to supply 20 g

of fat, 30 g of carbohydrates, and 44 g of protein?

84. Training diet: A physical trainer is designing a workout diet for one

of her clients, and wants to supply him with 24 g of fat, 244 g of

carbohydrates, and 40 g of protein for the noontime meal. The table

given shows three different foods and the amount of these nutrients (in

grams) that each ounce of food provides. How many ounces of each

should the trainer recommend?







Nutrient



Food I



Food II



Food III



Fat



2



4



3



Carb.



4



2



5



Protein



5



6



7



Food I



Food II



Food III



Fat



2



5



0



Carb.



10



15



18



Protein



2



10



0.75



Nutrient



EXTENDING THE CONCEPT



85. Some matrix applications require that you solve a matrix equation of the form AX ϩ B ϭ C, where A, B, and C

are matrices with the appropriate number of rows and columns and AϪ1 exists. Investigate the solution process

2

3

4

12

x

d, B ϭ c d, C ϭ c

d , and X ϭ c d , then solve AX ϩ B ϭ C for X

for such equations using A ϭ c

Ϫ5 Ϫ4

9

Ϫ4

y

Ϫ1

symbolically (using A , I, and so on).

86. Another alternative for finding determinants uses the triangularized form of a matrix and is offered without

proof: If nonsingular matrix A is written in triangularized form using standard row operations but without

exchanging any rows and without using the operation kRi to replace any row (k a constant), then det(A) is

equal to the product of resulting diagonal entries. Compute the determinant of each matrix using this method.

Be careful not to interchange rows and do not replace any row by a multiple of that row in the process.

1

a. £ Ϫ4

2







Ϫ2 3

5 Ϫ6 §

5

3



2

b. £ Ϫ2

4



5 Ϫ1

Ϫ3 4 §

6

5



Ϫ2 4

c. £ 5

7

3 Ϫ8



1

Ϫ2 §

Ϫ1



3

d. £ 0

Ϫ2



Ϫ1 4

Ϫ2 6 §

1 Ϫ3



MAINTAINING YOUR SKILLS



87. (4.2) Solve using the rational zeroes theorem:

x3 Ϫ 7x2 ϭ Ϫ36



89. (2.3) Solve the absolute value inequality:

Ϫ3Ϳ2x ϩ 5Ϳ Ϫ 7 Յ Ϫ19.



88. (2.2/5.3) Match each equation to its related graph.

Justify your answers.

y ϭ log2 x Ϫ 2

y ϭ log2 1x Ϫ 22

a. y

b.

y



90. (2.6) A coin collector believes that the value of

a coin varies inversely as the number of coins

still in circulation. If 4 million coins are in

circulation, the coin has a value of $25. (a) Find

the variation equation and (b) determine how

many coins are in circulation if the value of the

coin is $6.25.



5

4

3

2

1



5

4

3

2

1

Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

Ϫ6



1 2 3 4 5 6 7 8 9 10 x



Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

Ϫ6



xϭ2



1 2 3 4 5 6 7 8 9 10 x



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7.4



Applications of Matrices and Determinants:

Cramer’s Rule, Partial Fractions, and More



LEARNING OBJECTIVES

In Section 7.4 you will see

how we can:



A. Solve a system using

determinants and

Cramer’s rule

B. Decompose a rational

expression into partial

fractions

C. Use determinants in

applications involving

geometry in the

coordinate plane



In addition to solving systems, matrices can be used to accomplish such diverse things as

finding the volume of a three-dimensional solid or establishing certain geometrical relationships in the coordinate plane. Numerous uses are also found in higher mathematics,

such as checking whether solutions to a differential equation are linearly independent.



A. Solving Systems Using Determinants and Cramer’s Rule

In addition to identifying singular matrices, determinants can actually be used to

develop a formula for the solution of a system. Consider the following solution to a

general 2 ϫ 2 system, which parallels the solution to a specific 2 ϫ 2 system. With a

view toward a solution involving determinants, the coefficients of x are written as a11

and a21 in the general system, and the coefficients of y are a12 and a22.

Specific System

e



General System



2x ϩ 5y ϭ 9

3x ϩ 4y ϭ 10



e



eliminate the x-term

Ϫ3R1 ϩ 2R2



a11x ϩ a12y ϭ c1

a21x ϩ a22y ϭ c2

eliminate the x-term

Ϫa21R1 + a11R2



sums to zero



sums to zero



Ϫ3 # 2x Ϫ 3 # 5y ϭ Ϫ3 # 9

Ϫa21a11x Ϫ a21a12y ϭ Ϫa21c1

e

e

2 # 3x ϩ 2 # 4y ϭ 2 # 10

a11a21x ϩ a11a22y ϭ a11c2

#

#

#

#

2 4y Ϫ 3 5y ϭ 2 10 Ϫ 3 9

a11a22y Ϫ a21a12y ϭ a11c2 Ϫ a21c1

Notice the x-terms sum to zero in both systems. We are deliberately leaving the solution on the left unsimplified to show the pattern developing on the right. Next we

solve for y.

Factor Out y



12 # 4 Ϫ 3 # 52y ϭ 2 # 10 Ϫ 3 # 9

2 # 10 Ϫ 3 # 9

yϭ #

2 4Ϫ3#5



Factor Out y



divide



1a11a22 Ϫ a21a12 2y ϭ a11c2 Ϫ a21c1

a11c2 Ϫ a21c1

divide y ϭ

a11a22 Ϫ a21a12



On the left we find y ϭ Ϫ7

Ϫ7 ϭ 1 and back-substitution shows x ϭ 2. But more

important, on the right we obtain a formula for the y-value of any 2 ϫ 2 system:

a11c2 Ϫ a21c1



. If we had chosen to solve for x, the solution would be

a11a22 Ϫ a21a12

a22c1 Ϫ a12c2

. Note these formulas are defined only if a11a22 Ϫ a21a12 0.



a11a22 Ϫ a21a12

You may have already noticed, but this denominator is the determinant of the matrix of

a11 a12

coefficients c

d from the previous section! Since the numerator is also a difference

a21 a22

of two products, we investigate the possibility that it too can be expressed as a determinant. Working backward, we’re able to reconstruct the numerator for x in determinant

c1 a12

d , where it is apparent this matrix was formed by replacing the coefform as c

c2 a22

ficients of the x-variables with the constant terms.

(removed)



a11 a12

`a

a22 `

21



remove

coefficients of x



7–43



`



a12

a22 `



c1

`c

2



a12

a22 `



replace

with constants



679



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