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
2§
1
0
2
£1
3
3
Ϫ4
1
Ϫ1
2§
0
2
£1
3
Ϫ1
2§
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
f§
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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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
3§
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
2§
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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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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College Algebra G&M—
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
yϭ
. 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.
xϭ
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