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136 Chapter 3 Determinants
3.4 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Finding the Adjoint and Inverse of a Matrix In
Exercises 1–8, find the adjoint of the matrix Then use
the adjoint to find the inverse of (if possible).
1. 2.
3. 4.
5. 6.
7.
8.
9. Proof Prove that if and all entries of are
integers, then all entries of must also be integers.
10. Proof Prove that if an matrix is not invertible,
then is the zero matrix.
Proof In Exercises 11 and 12, prove the formula for a
nonsingular matrix Assume
11. 12.
13. Illustrate the formula in Exercise 11 using the matrix
14. Illustrate the formula in Exercise 12 using the matrix
15. Proof Prove that if is an invertible matrix,
then
16. Illustrate the formula in Exercise 15 using the matrix
Using Cramer’s Rule In Exercises 17–30, use Cramer’s
Rule to solve (if possible) the system of linear equations.
17. 18.
19. 20.
21. 22.
23. 24.
25.
26.
27.
28.
29.
30.
Using Cramer’s Rule In Exercises 31–34, use a
software program or a graphing utility with matrix
capabilities and Cramer’s Rule to solve (if possible) the
system of linear equations.
31.
32.
33.
34.
35. Use Cramer’s Rule to solve the system of linear
equations for and
For what value(s) of will the system be inconsistent?k
s1 2 kdx 1 ky 5 3
kx 1 s1 2 kdy 5 1
y.x
2x1
3x1
22x1
2
1
2
x2
5x2
3x2
1
2
5x3
2x3
3x3
1
1
x4
x4
5
5
5
5
28
24
26
215
3x1
2x1
2x1
2
1
2x2
2x2
1
2
9x3
9x3
3x3
1
2
1
1
4x4
6x4
x4
8x4
5
5
5
5
35
217
5
24
28x1
12x1
15x1
1
1
2
7x2
3x2
9x2
2
2
1
10x3
5x3
2x3
5
5
5
2151
86
187
56x143x1
2
2
x272x2
5
5
220
251
2x1
3x1
5x1
1
1
1
3x2
5x2
9x2
1
1
1
5x3
9x3
17x3
5
5
5
4
7
13
4x1
2x1
5x1
2
1
2
x2
2x2
2x2
1
1
1
x3
3x3
6x3
5
5
5
25
10
1
14x1
24x1
56x1
2
1
2
21x2
2x2
21x2
2
2
1
7x3
2x3
7x3
5
5
5
221
2
7
3x1
4x1
6x1
1
2
2
4x2
4x2
6x2
1
1
4x3
6x3
5
5
5
11
11
3
4x1
2x1
8x1
2
1
2
2x2
2x2
5x2
1
1
2
3x3
5x3
2x3
5
5
5
22
16
4
4x1
2x1
5x1
2
1
2
x2
2x2
2x2
2
1
2
x3
3x3
2x3
5
5
5
1
10
21
2x1 2 4x2 5 5.0
20.4x1
0.2x1
1
1
0.8x2
0.3x2
5
5
1.6
0.6
20.4x1 1 0.8x2 5 1.6
26x1 2 12x2 5 8 12x1 2 24x2 5 21
13x1 2 6x2 5 17 20x1 1 8x2 5 11
30x1 1 24x2 5 23 5x1 1 3x2 5 4
18x1 1 12x2 5 13 3x1 1 4x2 5 22
3x1 1 2x2 5 21 2x1 1 x2 5 1
2x1 2 x2 5 210 x1 1 2x2 5 5
A 5 31
1
3
24.
fadjsAdg21.adjsA21d 5n 3 nA
A 5 321
1
3
24.
A 5 31
1
0
224.
adjfadjsAdg 5 |A|n22A|adjsAd| 5 |A|n21
n $ 2.A.n 3 n
AfadjsAdgAn 3 n
|A21|A|A| 5 1
A 5 31
1
1
0
1
1
0
1
1
0
1
1
0
1
1
14
A 5 321
3
0
21
2
21
0
1
0
4
1
1
1
1
2
24
A 5 30
1
21
1
2
21
1
3
224A 5 3
23
2
0
25
4
1
27
3
214
A 5 31
0
2
2
1
2
3
21
24A 5 3
1
0
0
0
2
24
0
6
2124
A 5 321
0
0
44A 5 31
3
2
44A
A.
3.4 Exercises 137
36. Verify the following system of linear equations in
and for the triangle shown in the figure.
Then use Cramer’s Rule to solve for and use
the result to verify the Law of Cosines,
Finding the Area of a Triangle In Exercises 37–40, find
the area of the triangle with the given vertices.
37. 38.
39. 40.
Testing for Collinear Points In Exercises 41–44,
determine whether the points are collinear.
41. 42.
43.
44.
Finding an Equation of a Line In Exercises 45–48,
find an equation of the line passing through the given
points.
45. 46.
47. 48.
Finding the Volume of a Tetrahedron In Exercises
49–52, find the volume of the tetrahedron with the given
vertices.
49.
50.
51.
52.
Testing for Coplanar Points In Exercises 53–56,
determine whether the points are coplanar.
53.
54.
55.
56.
Finding an Equation of a Plane In Exercises 57–60,
find an equation of the plane passing through the given
points.
57.
58.
59.
60.
Using Cramer’s Rule In Exercises 61 and 62,
determine whether Cramer’s Rule is used correctly to
solve for the variable. If not, identify the mistake.
61.
62.
63. Textbook Publishing The table shows the estimated
revenues (in millions of dollars) of textbook publishers
in the United States from 2007 through 2009. (Source:
U.S. Census Bureau)
(a) Create a system of linear equations for the data to fit
the curve
where corresponds to 2007, and is the
revenue.
(b) Use Cramer’s Rule to solve the system.
(c) Use a graphing utility to plot the data and graph
the polynomial function in the same viewing
window.
(d) Briefly describe how well the polynomial function
fits the data.
yt 5 7
y 5 at 2 1 bt 1 c
Year, t Revenues, y
2007 10,697
2008 11,162
2009 9891
x 5| 15
27
23
22
23
21
1
21
27||532 22
23
21
1
21
27|5x 2
3x 2
2x 2
2y 1
3y 2
y 2
z 5
z 5
7z 5
15
27
23
y 5| 1
21
4
2
3
1
1
22
21|| 1
21
4
2
4
6
1
22
21|x 1
2x 1
4x 1
2y 1
3y 2
y 2
z 5 2
2z 5 4
z 5 6
s1, 2, 7d, s4, 4, 2d, s3, 3, 4ds0, 0, 0d, s1, 21, 0d, s0, 1, 21d
s0, 21, 0d, s1, 1, 0d, s2, 1, 2ds1, 22, 1d, s21, 21, 7d, s2, 21, 3d
s1, 2, 7d, s23, 6, 6d, s4, 4, 2d, s3, 3, 4ds0, 0, 21d, s0, 21, 0d, s1, 1, 0d, s2, 1, 2ds1, 2, 3d, s21, 0, 1d, s0, 22, 25d, s2, 6, 11ds24, 1, 0d, s0, 1, 2d, s4, 3, 21d, s0, 0, 1d
s0, 0, 0d, s0, 2, 0d, s3, 0, 0d, s1, 1, 4ds3, 21, 1d, s4, 24, 4d, s1, 1, 1d, s0, 0, 1ds1, 1, 1d, s0, 0, 0d, s2, 1, 21d, s21, 1, 2ds1, 0, 0d, s0, 1, 0d, s0, 0, 1d, s1, 1, 1d
s1, 4d, s3, 4ds22, 3d, s22, 24ds24, 7d, s2, 4ds0, 0d, s3, 4d
s21, 23d, s24, 7d, s2, 213ds22, 5d, s0, 21d, s3, 29d
s21, 0d, s1, 1d, s3, 3ds1, 2d, s3, 4d, s5, 6d
s1, 1d, s21, 1d, s0, 22ds21, 2d, s2, 2d, s22, 4ds1, 1d, s2, 4d, s4, 2ds0, 0d, s2, 0d, s0, 3d
A
ab
c
C
B
c2 5 a2 1 b2 2 2ab cos C.
cos C,
b cos A 1 a cos B 5 c
c cos A 1 a cos C 5 b
c cos B 1 b cos C 5 a
cos Ccos B,
cos A,
64. Consider the system of linear
equations
where and represent real
numbers. What must be true about the lines
represented by the equations when
|a1
a2
b1
b2| 5 0?
c2b2,a2,c1,b1,a1,
5a1x
a2x
1
1
b1y
b2y
5
5
c1
c2
138 Chapter 3 Determinants
The Determinant of a Matrix In Exercises 1–18, find
the determinant of the matrix.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15.
16.
17.
18.
Properties of Determinants In Exercises 19–22,
determine which property of determinants the equation
illustrates.
19.
20.
21.
22.
The Determinant of a Matrix Product In Exercises 23
and 24, find (a) (b) (c) and (d) Then
verify that
23.
24.
Finding Determinants In Exercises 25 and 26, find
(a) (b) (c) and (d)
25. 26.
Finding Determinants In Exercises 27 and 28, find
(a) and (b)
27. 28.
The Determinant of the Inverse of a Matrix In
Exercises 29–32, find Begin by finding
and then evaluate its determinant. Verify your result
by finding and then applying the formula from
Theorem 3.8,
29. 30.
31. 32. 321
2
1
1
4
21
2
8
043
1
2
2
0
21
6
1
4
04
3 10
22
2
7431
2
21
44
|A21| 5
1
|A|.
|A|
A21,|A21|.
A 5 32
5
1
21
0
22
4
3
04A 5 3
1
0
22
0
3
7
24
2
64
|A21|.|A|
A 5 33
21
2
0
0
1
1
0
24A 5 322
1
6
34
|5A|.|ATA|,|A3|,|AT|,
B 5 31
0
0
2
21
2
1
1
34A 5 3
1
4
7
2
5
8
3
6
04,
B 5 33
2
4
14A 5 321
0
2
14,
|A||B| 5 |AB|.|AB|.AB,|B|,|A|,
|101 3
21
2
1
2
1| 5 |121 3
5
2
1
4
1||2016 24
4
8
12
3
6
9
26
2
1
0
1| 5 212|2016 1
21
22
23
1
2
3
22
2
1
0
1||124 2
0
21
21
3
1| 5 2 |124 21
3
1
2
0
21||26 21
23| 5 0
30
0
0
0
2
0
0
0
2
0
0
0
2
0
0
0
2
0
0
0
2
0
0
0
0
43
21
0
0
0
0
0
21
0
0
0
0
0
21
0
0
0
0
0
21
0
0
0
0
0
21
43
1
2
1
1
0
2
3
2
0
21
21
21
0
2
1
3
2
1
21
0
4
22
21
0
2
43
21
0
1
0
0
1
1
0
21
1
21
21
1
0
1
0
0
21
1
21
0
1
0
21
1
43
3
22
21
22
21
0
2
1
2
1
23
22
1
23
4
143
24
1
2
1
1
22
21
2
2
1
3
2
3
2
4
214
32
23
4
5
0
1
21
2
0
0
3
1
0
0
0
2143
2
21
3
22
0
2
0
0
21
0
1
3
4
3
2
14
3215
3
12
0
9
23
3
26
643
23
9
0
6
12
15
9
23
264
3215
3
12
0
0
0
4
25
643
22
0
0
0
23
0
0
0
214
35
0
0
0
21
0
2
3
143
1
0
1
4
23
1
22
1
214
322
0
0
34323
6
1
224
30
1
23
2434
2
21
24
3 Review Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Review Exercises 139
Solving a System of Linear Equations In Exercises
33–36, solve the system of linear equations by each of the
following methods.
(a) Gaussian elimination with back-substitution
(b) Gauss-Jordan elimination
(c) Cramer’s Rule
33.
34.
35.
36.
System of Linear Equations In Exercises 37–42, use
the determinant of the coefficient matrix to determine
whether the system of linear equations has a unique
solution.
37. 38.
39. 40.
41.
42.
43. Let and be square matrices of order 4 such that
and Find (a) (b) (c)
(d) and (e)
44. Let and be square matrices of order 3 such that
and Find (a) (b) (c)
(d) and (e)
45. Proof Prove the following property.
46. Illustrate the property in Exercise 45 with the following.
47. Find the determinant of the matrix.
48. Show that
Calculus In Exercises 49–52, find the Jacobians of
the functions. If and are continuous functions
of and with continuous first partial derivatives,
then the Jacobians and are defined as
and
49.
50.
51.
52.
53. Writing Compare the various methods for calculating
the determinant of a matrix. Which method requires the
least amount of computation? Which method do you
prefer when the matrix has very few zeros?
54. Writing A computer operator charges $0.001 (one
tenth of a cent) for each addition and subtraction, and
$0.003 for each multiplication and division. Use the
table on page 116 to compare and contrast the costs
of calculating the determinant of a matrix
by cofactor expansion and then by row reduction.
Which method would you prefer to use for calculating
determinants?
55. Writing Solve the equation for if possible. Explain
your result.
56. Proof Prove that if and and are of
the same size, then there exists a matrix such that
and A 5 CB.|C| 5 1
C
BA|A| 5 |B| Þ 0,
| cos x
sin x
sin x 2 cos x
0
0
1
sin x
cos x
sin x 1 cos x
| 5 0
x,
10 3 10
z 5 u 1 v 1 wy 5 2uv,x 5 u 2 v 1 w,
z 5 2uvwy 512 su 2 vd,x 5
12 su 1 vd,
y 5 cu 1 dvx 5 au 1 bv,
y 512 sv 1 udx 5
12sv 2 ud,
Jxu, v, wc 5 |x
u
y
u
z
u
x
v
y
v
z
v
x
w
y
w
z
w|.Jxu, vc 5 |xu
y
u
xv
y
v|Jxu, v, wcJxu, vc
wu, v,
zx, y,
|a111 1
a
1
1
1
1
a
1
1
1
1
a| 5 sa 1 3dsa 2 1d3.
31 2 n
1...1
1
1 2 n...1
1
1...1
. . .
. . .
. . .
1
1...
1 2 n4
n 3 n
c33 5 1c32 5 0,c31 5 3,A 5 31
1
2
0
21
1
2
2
214,
|a11
a21
a31
a12
a22
a32
a13
a23
a33| 1 |a11
a21
c31
a12
a22
c32
a13
a23
c33|| a11
a21
a31 1 c31
a12
a22
a32 1 c32
a13
a23
a33 1 c33| 5
|B21|.|sABdT|,|2A|,|B4|,|BA|,|B| 5 5.|A| 5 22
BA
|B21|.|sABdT|,|2A|,|B2|,|BA|,|B| 5 2.|A| 5 4
BA
2x1 2 x3 5 0
2x1 1 4x2 2 2x5 5 0
3x3 1 8x4 1 6x5 5 16
4x1 1 2x2 1 5x3 5 3
x1 1 5x2 1 3x3 5 14
3x1 1 x2 1 3x3 5 26
2x1 1 5x2 1 15x3 5 4
x1 1 2x2 1 6x3 5 1
8x 1 6y 5 22 5x 1 4y 1 2z 5 4
2x 2 3y 2 3z 5 22 2x 1 3y 1 z 5 22
2x 1 3y 1 z 5 10 2x 1 y 1 2z 5 1
3x 2 7y 5 1 2x 1 y 5 222
2x 2 5y 5 2 5x 1 4y 5 2
2x1
3x1
5x1
1
1
1
3x2
5x2
9x2
1
1
1
5x3
9x3
13x3
5
5
5
4
7
17
x1
2x1
2x1
1
2
1
2x2
2x2
3x2
2
2
1
x3
2x3
4x3
5
5
5
27
28
8
2x1
2x1
3x1
1
1
1
x2
2x2
2x2
1
2
2
2x3
3x3
x3
5
5
5
6
0
6
3x1
3x1
5x1
1
1
1
3x2
5x2
9x2
1
1
1
5x3
9x3
17x3
5
5
5
1
2
4
140 Chapter 3 Determinants
Finding the Adjoint of a Matrix In Exercises 57 and
58, find the adjoint of the matrix.
57. 58.
System of Linear Equations In Exercises 59–62, use
the determinant of the coefficient matrix to determine
whether the system of linear equations has a unique
solution. If it does, use Cramer’s Rule to find the solution.
59. 60.
61.
62.
Using Cramer’s Rule In Exercises 63 and 64, use a
software program or a graphing utility with matrix
capabilities and Cramer’s Rule to solve (if possible) the
system of linear equations.
63.
64.
Finding the Area of a Triangle In Exercises 65 and 66,
use a determinant to find the area of the triangle with the
given vertices.
65. 66.
Finding an Equation of a Line In Exercises 67 and 68,
use a determinant to find an equation of the line passing
through the given points.
67. 68.
Finding an Equation of a Plane In Exercises 69 and
70, use a determinant to find an equation of the plane
passing through the given points.
69.
70.
71. Using Cramer’s Rule Determine whether Cramer’s
Rule is used correctly to solve for the variable. If not,
identify the mistake.
72. Health Care Expenditures The table shows annual
personal health care expenditures (in billions of dollars)
in the United States from 2007 through 2009. (Source:
Bureau of Economic Analysis)
(a) Create a system of linear equations for the data to fit
the curve
where corresponds to 2007, and is the
amount of the expenditure.
(b) Use Cramer’s Rule to solve the system.
(c) Use a graphing utility to plot the data and graph the
polynomial function in the same viewing window.
(d) Briefly describe how well the polynomial function
fits the data.
True or False? In Exercises 73–76, determine whether
each statement is true or false. If a statement is true, give
a reason or cite an appropriate statement from the text.
If a statement is false, provide an example that shows the
statement is not true in all cases or cite an appropriate
statement from the text.
73. (a) The cofactor of a given matrix is always a
positive number.
(b) If a square matrix is obtained from by
interchanging two rows, then
(c) If one column of a square matrix is a multiple of
another column, then the determinant is 0.
(d) If is a square matrix of order then
74. (a) If and are square matrices of order such that
then both and are nonsingular.
(b) If is a matrix with then
(c) If and are square matrices of order then
75. (a) In Cramer’s Rule, the value of is the quotient
of two determinants, where the numerator is the
determinant of the coefficient matrix.
(b) Three points and are
collinear when the determinant of the matrix that has
the coordinates as entries in the first two columns
and 1’s as entries in the third column is nonzero.
76. (a) If is a square matrix, then the matrix of cofactors
of is called the adjoint of
(b) In Cramer’s Rule, the denominator is the determinant
of the matrix formed by replacing the column
corresponding to the variable being solved for with
the column representing the constants.
A.A
A
sx3, y3dsx2, y2d,sx1, y1d,
xi
detsAd 1 detsBd.detsA 1 Bd 5
n,BA
dets2Ad 5 10.
detsAd 5 5,3 3 3A
BAdetsABd 5 21,
nBA
2detsAT d.detsAd 5
n,A
detsBd 5 detsAd.AB
C22
yt 5 7
y 5 at 2 1 bt 1 c
Year, t 2007 2008 2009
Amount, y 1465 1547 1623
z 5|21
6
1
24
23
1
21
1
24||121 24
23
1
21
1
24|x 2
2x 2
x 1
4y 2
3y 1
y 2
z 5
z 5
4z 5
21
6
1
s0, 0, 0d, s2, 21, 1d, s23, 2, 5ds0, 0, 0d, s1, 0, 3d, s0, 3, 4d
s2, 5d, s6, 21ds24, 0d, s4, 4d
s24, 0d, s4, 0d, s0, 6ds1, 0d, s5, 0d, s5, 8d
4x1
2x1
5x1
2
1
2
x2
2x2
2x2
1
1
1
x3
3x3
6x3
5
5
5
25
10
1
0.2x1
2x1
2
1
0.6x2
1.4x2
5
5
2.4
28.8
8x1 1 2x2 2 4x3 5 6
4x1 2 2x2 2 8x3 5 1
4x1 1 4x2 1 4x3 5 5
12x1 1 9x2 2 x3 5 2
6x1 1 6x2 1 12x3 5 13
2x1 1 3x2 1 3x3 5 3
3x 2 y 5 21.3 0.4x 2 0.5y 5 20.01
2x 1 y 5 0.3 0.2x 2 0.1y 5 0.07
31
0
0
21
1
0
1
2
2143 0
22
1
14