144 Chapter 3 Determinants

3 Review Exercises See CalcChat.com for worked-out solutions to odd-numbered exercises.

The Determinant of a Matrix In Exercises 1–18, find Properties of Determinants In Exercises 19–22,
the determinant of the matrix. determine which property of determinants the equation
illustrates.
[42 −12] [01 −32]
∣ ∣
1. 2.
4 −1
19. =0
−4

∣ ∣ ∣ ∣
−3 −2 16
3. [
6 −2]
4. [
3]
1 0
0 1 2 −1 1 −1 2
−1 −4 3 =− 2

[ ] [ ]
3 5 0 2 20. 2 0 3 0

∣ ∣ ∣ ∣
5. 0 −2 −1 6. 0 −1 3 4 −1 1 4 1 −1
−1 −1 1 0 0 1 2 −4 3 2 2 1 1 2
−2 −15 −1

[ ] [ ]
0 0 0 4 0 4 6 1 0 2 1
21. = −12
7. 0 −3 0 8. 3 0 −5 1 8 9 0 1 −2 3 0
−1 −6 −3 −2

∣ ∣∣ ∣
0 0 12 0 6 6 12 1 6 1
−3 −15

[ ] [ ]
6 9 0 3 1 3 1 1 3 1
9. 9 12 −3 10. 3 9 −6 22. 0 −1 2 = 2 5 4
0 15 −6 12 −3 6 1 2 1 1 2 1

[ ] [ ]
2 0 −1 4 2 0 0 0 The Determinant of a Matrix Product In Exercises 23
−1 −3
11.
2 0 3
12.
1 0 0
∣ ∣ ∣ ∣
and 24, find (a) A , (b) B , (c) AB, and (d) AB . Then ∣ ∣
3
−2
0
0
1
3
2
1
4
5
−1
2
3
1
0
−1
verify that A B = AB . ∣ ∣∣ ∣ ∣ ∣
−1
[ ] [ ]
2 3 4
23. A = , B=

[ ] [ ]
−4 1 2 3 3 −1 2 1 0 1 2 1
1 −2 1 2 −2 0 1 −3

[ ] [ ]
13. 14. 0 1 2 2 1 2
2 −1 3 4 −1 2 −3 4 24. A = 5 4 3 , B= 1 −1 0
1 2 2 −1 −2 1 −2 1 7 6 8 0 3 −2

[ ]
−1 1 −1 0 0 Finding Determinants In Exercises 25 and 26, find
−1
15.
0
1
1
0 1
0
−1
1
0
∣ ∣ ∣ ∣
(a) AT , (b) A3 , (c) ATA , and (d) 5A . ∣ ∣ ∣ ∣

[ ]
3 0 1
−3
−1 −1
[ ]
0 0 1 8
25. A = 26. A = −1 0 0
0 1 1 −1 1 4 1
2 1 2

[ ]
1 2 −1 3 4
Finding Determinants In Exercises 27 and 28, find
−1 −2
16.
2
1
3
2 0
2
1 −1
∣ ∣
(a) A and (b) A−1 . ∣ ∣
−4 −2

[ ] [ ]
1 0 1 3
1 0 2 −1 0
27. A = 0 3 2 28. A = 2 0 4
0 −1 1 0 2
−2 7 6 −1 5 0

[ ]
−1 0 0 0 0
The Determinant of the Inverse of a Matrix In
0 −1 0 0 0
17. 0 0 −1 0 0
Exercises 29–32, find A−1 . Begin by finding A−1, ∣ ∣
and then evaluate its determinant. Verify your result
−1
0
0
0
0
0
0 0
0
−1
∣ ∣
by finding A and then applying the formula from
1
Theorem 3.8, A−1 =
A
. ∣ ∣ ∣ ∣

[ ]
0 0 0 0 3

[−21 ] [−2 ]
0 0 0 3 0 4 10 2
29. A = 30. A =
18. 0 0 3 0 0 1 7
−1

[ ] [ ]
0 3 0 0 0 1 0 1 1 2
3 0 0 0 0 31. A = 2 −1 4 32. A = 2 4 8
2 6 0 1 −1 0

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 Review Exercises 145

Solving a System of Linear Equations In Exercises 46. Illustrate the property in Exercise 45 with A, c31, c32,
33–36, solve the system of linear equations by each of the and c33 below.
methods listed below.

[ ]
1 0 2
(a) Gaussian elimination with back-substitution A = 1 −1 2 , c31 = 3, c32 = 0, c33 = 1
(b) Gauss-Jordan elimination 2 1 −1
(c) Cramer’s Rule 47. Find the determinant of the n × n matrix.

[ ]
33. 3x1 + 3x2 + 5x3 = 1 1−n 1 1 . . . 1
3x1 + 5x2 + 9x3 = 2 1 1−n 1 . . . 1
5x1 + 9x2 + 17x3 = 4 ⋮ ⋮ ⋮ ⋮
1 1 1 . . . 1−n
34. x1 + 2x2 + x3 = 4

∣ ∣
−3x1 + x2 − 2x3 = 1 48. Show that
2x1 + 3x2 − x3 = 9 a 1 1 1
35. x1 + 2x2 − x3 = −7 1 a 1 1
= (a + 3)(a − 1)3.
2x1 − 2x2 − 2x3 = −8 1 1 a 1
−x1 + 3x2 + 4x3 = 8 1 1 1 a
36. 2x1 + 3x2 + 5x3 = 4 Calculus In Exercises 49–54, find the Jacobians of the
3x1 + 5x2 + 9x3 = 7 functions. If x, y, and z are continuous functions of u, v,

∣ ∣
5x1 + 9x2 + 13x3 = 17 and w with continuous first partial derivatives, then the
Jacobians J(u, v) and J(u, v, w) are

∣ ∣
System of Linear Equations In Exercises 37–42, use
the determinant of the coefficient matrix to determine 𝛛x 𝛛x 𝛛x
whether the system of linear equations has a unique 𝛛x 𝛛x 𝛛u 𝛛v 𝛛w
solution. 𝛛u 𝛛v 𝛛y 𝛛y 𝛛y
J(u, v) = and J(u, v, w) = .
37. 6x + 5y = 0 38. 2x − 5y = 2 𝛛y 𝛛y 𝛛u 𝛛v 𝛛w
x − y = 22 3x − 7y = 1 𝛛u 𝛛v 𝛛z 𝛛z 𝛛z
𝛛u 𝛛v 𝛛w
39. −x + y + 2z = 1 40. 2x + 3y + z = 10
2x + 3y + z = −2 2x − 3y − 3z = 22 49. x = 21(v − u), y = 12(v + u)
5x + 4y + 2z = 4 8x + 6y = −2 50. x = au + bv, y = cu + dv
41. x1 + 2x2 + 6x3 = 1 51. x = u cos v, y = u sin v
2x1 + 5x2 + 15x3 = 4 52. x = eu sin v, y = eu cos v
3x1 + x2 + 3x3 = −6 53. x = 21(u + v), y = 12(u − v), z = 2uvw
42. x1 + 5x2 + 3x3 = 14 54. x = u − v + w, y = 2uv, z = u + v + w
4x1 + 2x2 + 5x3 = 3
3x3 + 8x4 + 6x5 = 16 55. Writing Compare the various methods for calculating
2x1 + 4x2 − 2x5 = 0 the determinant of a matrix. Which method requires the
2x1 − x3 = 0 least amount of computation? Which method do you
prefer when the matrix has very few zeros?
43. Let A and B be square matrices of order 4 such that 56. Writing Use the table on page 122 to compare the
∣∣ ∣∣
A = 4 and B = 2. Find (a) BA , (b) B2 , (c) 2A , ∣ ∣ ∣ ∣ ∣ ∣ numbers of operations involved in calculating the
∣ ∣
(d) (AB)T , and (e) B −1 . ∣ ∣ determinant of a 10 × 10 matrix by cofactor expansion
44. Let A and B be square matrices of order 3 such that and then by row reduction. Which method would you
∣∣ ∣∣
A = −2 and B = 5. Find (a) BA , (b) B 4 , (c) 2A , ∣ ∣ ∣ ∣ ∣ ∣ prefer to use for calculating determinants?
∣ ∣
(d) (AB)T , and (e) B −1 . ∣ ∣ 57. Writing Solve the equation for x, if possible.

∣ ∣∣ ∣ ∣ ∣
45. Proof Prove the property below. cos x 0 sin x
a11 a12 a13 a11 a12 a13 sin x 0 cos x = 0
a21 a22 a23 = a21 a22 a23 sin x − cos x 1 sin x + cos x

∣ ∣
a31 + c31 a32 + c32 a33 + c33 a31 a32 a33
a11 a12 a13
58. Proof Prove that if A = B ≠ 0, and A and B are of∣∣ ∣∣
the same size, then there exists a matrix C such that
+ a21 a22 a23
c31 c32 c33 ∣C∣ = 1 and A = CB.

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
146 Chapter 3 Determinants

Finding the Adjoint of a Matrix In Exercises 59 and 74. Health Care Expenditures The table shows annual
60, find the adjoint of the matrix. personal health care expenditures (in billions of dollars)
−1 in the United States from 2011 through 2013. (Source:

[ ]
1 1
[ ]
0 1 Bureau of Economic Analysis)
59. 60. 0 1 2
−2 1
0 0 −1
Year 2011 2012 2013
System of Linear Equations In Exercises 61–64, use
Amount, y 1765 1855 1920
the determinant of the coefficient matrix to determine
whether the system of linear equations has a unique
(a) Create a system of linear equations for the data to fit
solution. If it does, use Cramer’s Rule to find the solution.
the curve
61. 0.2x − 0.1y = 0.07 62. 2x + y = 0.3
y = at2 + bt + c
0.4x − 0.5y = −0.01 3x − y = −1.3
where t = 1 corresponds to 2011, and y is the
63. 2x1 + 3x2 + 3x3 = 3
amount of the expenditure.
6x1 + 6x2 + 12x3 = 13
(b) Use Cramer’s Rule to solve the system.
12x1 + 9x2 − x3 = 2
(c) Use a graphing utility to plot the data and graph the
64. 4x1 + 4x2 + 4x3 = 5
polynomial function in the same viewing window.
4x1 − 2x2 − 8x3 = 1
8x1 + 2x2 − 4x3 = 6 (d) Briefly describe how well the polynomial function
fits the data.
Using Cramer’s Rule In Exercises 65 and 66, use a True or False? In Exercises 75–78, determine whether
software program or a graphing utility and Cramer’s each statement is true or false. If a statement is true, give
Rule to solve (if possible) the system of linear equations. a reason or cite an appropriate statement from the text.
65. 0.2x1 − 0.6x2 = 2.4 If a statement is false, provide an example that shows the
−x1 + 1.4x2 = −8.8 statement is not true in all cases or cite an appropriate
66. 4x1 − x2 + x3 = −5 statement from the text.
2x1 + 2x2 + 3x3 = 10 75. (a) The cofactor C22 of a matrix is always a positive
5x1 − 2x2 + 6x3 = 1 number.
(b) If a square matrix B is obtained from A by
Finding the Area of a Triangle In Exercises 67 and 68,
interchanging two rows, then det(B) = det(A).
use a determinant to find the area of the triangle with
the given vertices. (c) If one column of a square matrix is a multiple of
another column, then the determinant is 0.
67. (1, 0), (5, 0), (5, 8) 68. (−4, 0), (4, 0), (0, 6)
(d) If A is a square matrix of order n, then
Finding an Equation of a Line In Exercises 69 and 70, det(A) = −det(AT ).
use a determinant to find an equation of the line passing
76. (a) If A and B are square matrices of order n such that
through the points.
det(AB) = −1, then both A and B are nonsingular.
69. (−4, 0), (4, 4) 70. (2, 5), (6, −1)
(b) If A is a 3 × 3 matrix with det(A) = 5, then
Finding an Equation of a Plane In Exercises 71 and det(2A) = 10.
72, use a determinant to find an equation of the plane (c) If A and B are square matrices of order n, then
passing through the points. det(A + B) = det(A) + det(B).
71. (0, 0, 0), (1, 0, 3), (0, 3, 4) 77. (a) In Cramer’s Rule, the value of xi is the quotient
72. (0, 0, 0), (2, −1, 1), (−3, 2, 5) of two determinants, where the numerator is the
determinant of the coefficient matrix.
73. Using Cramer’s Rule Determine whether Cramer’s (b) Three points (x1, y1), (x2, y2), and (x3, y3 ) are
Rule is used correctly to solve for the variable. If not, collinear when the determinant of the matrix that has

∣ ∣
identify the mistake. the coordinates as entries in the first two columns
−1 −4 −1 and 1’s as entries in the third column is nonzero.
6 −3 1 78. (a) The matrix of cofactors of a square matrix A is the
x − 4y − z = −1

∣ ∣
1 1 −4 adjoint of A.
2x − 3y + z = 6 z=
1 −4 −1 (b) In Cramer’s Rule, the denominator is the determinant
x + y − 4z = 1
2 −3 1 of the matrix formed by replacing the column
1 1 −4 corresponding to the variable being solved for with
the column representing the constants.

Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.