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Deck 3: Determinants
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Question
Find the determinant of 
.
A) -14
B) 30
C) 22
D) -6
E) -18
.A) -14
B) 30
C) 22
D) -6
E) -18
Question
Evaluate the determinant. 
A) 16
B) -14
C) 8
D) 2
E) -2
A) 16
B) -14
C) 8
D) 2
E) -2
Question
Find the determinant of the matrix 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
Question
Find the minor 
and its cofactor 
of the matrix 
.
A)
B)
C)
D)
E)
and its cofactor of the matrix .A)
B)
C)
D)
E)
Question
Find the minor 
and its cofactor 
of the matrix 
.
A)
B)
C)
D)
E)
and its cofactor of the matrix .A)
B)
C)
D)
E)
Question
Find the determinant of 
by the method of expansion by cofactors.
A) -945
B) -702
C) 594
D) 378
E) 1,350
by the method of expansion by cofactors.A) -945
B) -702
C) 594
D) 378
E) 1,350
Question
Find the determinant of 
by the method of expansion by cofactors.
A) 192
B) 384
C) -48
D) -384
E) 176
by the method of expansion by cofactors.A) 192
B) 384
C) -48
D) -384
E) 176
Question
Use the matrix capabilities of a graphing utility to find the determinant of the matrix 
.
A) -0.98
B) -0.343
C) -1.715
D) 1.029
E) -0.686
.A) -0.98
B) -0.343
C) -1.715
D) 1.029
E) -0.686
Question
Use technology to find the determinant. 
A) 119
B) -119
C) -207
D) -115
E) 121
A) 119
B) -119
C) -207
D) -115
E) 121
Question
Solve for x given the following equation involving a determinant.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Solve for x given the following equation involving a determinant. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Evaluate the determinant of the matrix below to determine which of the following makes the equation true. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Which property of determinants is illustrated by the equation 
A) If every entry of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c.
B) If a single row or a single column of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c.
C) If a single row or a single column of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by 
.
D) If every entry of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by 
E) If c times one row is added to another row of an
matrix, the determinant of the matrix is multiplied by c.
A) If every entry of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c.B) If a single row or a single column of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c.C) If a single row or a single column of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by .D) If every entry of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by E) If c times one row is added to another row of an
matrix, the determinant of the matrix is multiplied by c. Question
Which property of determinants is illustrated by the equation 
A) If a multiple of a row is added to another row in a matrix and then the constant is added to a row, the determinant of the matrix is unchanged.
B) If a non-zero constant is added to a row of a matrix, the determinant of the matrix is unchanged.
C) If the two columns of the matrix are interchanged, the determinant of the matrix is unchanged.
D) If a multiple of a row is added to another row of a matrix, the determinant of the matrix is unchanged.
E) If times a row of a matrixis added to another row of a matrix, the determinant of the matrix is multiplied by c.
A) If a multiple of a row is added to another row in a matrix and then the constant is added to a row, the determinant of the matrix is unchanged.
B) If a non-zero constant is added to a row of a matrix, the determinant of the matrix is unchanged.
C) If the two columns of the matrix are interchanged, the determinant of the matrix is unchanged.
D) If a multiple of a row is added to another row of a matrix, the determinant of the matrix is unchanged.
E) If times a row of a matrixis added to another row of a matrix, the determinant of the matrix is multiplied by c.
Question
Which property of determinants is illustrated by the equation 
A) If two rows of a matrix are interchanged, the determinant of the matrix is multiplied by -1.
B) If two columns of a matrix are interchanged, the determinant of the matrix is multiplied by -1.
C) If a row of a matrix is subtracted from another row of the same matrix, the determinant of the matrix is multiplied by -1.
D) If two matrices have any rows equal, the determinant of the matrix is multiplied by -1.
E) If a column of a matrix is subtracted from another column, the determinant of the matrix is multiplied by -1.
A) If two rows of a matrix are interchanged, the determinant of the matrix is multiplied by -1.
B) If two columns of a matrix are interchanged, the determinant of the matrix is multiplied by -1.
C) If a row of a matrix is subtracted from another row of the same matrix, the determinant of the matrix is multiplied by -1.
D) If two matrices have any rows equal, the determinant of the matrix is multiplied by -1.
E) If a column of a matrix is subtracted from another column, the determinant of the matrix is multiplied by -1.
Question
Use the matrix capabilities of a graphing utility to find the determinant of the matrix 
.
A) -162
B) 0
C) -216
D) -188
E) 188
.A) -162
B) 0
C) -216
D) -188
E) 188
Question
Use elementary row or column operations to evaluate the determinant. 
A) 160
B) 0
C) -160
D) 120
E) -120
A) 160
B) 0
C) -160
D) 120
E) -120
Question
Use elementary row or column operations to evaluate the determinant. 
A) -84
B) 1,812
C) -1,212
D) 180
E) -180
A) -84
B) 1,812
C) -1,212
D) 180
E) -180
Question
Find the determinant of the elementary matrix 
.
A) -1
B) 1
C) -2
D) 0
E) 2
.A) -1
B) 1
C) -2
D) 0
E) 2
Question
Find the determinant of the elementary matrix 
.
A) 1
B) 1- k
C) k
D) -k
E) k - 1
.A) 1
B) 1- k
C) k
D) -k
E) k - 1
Question
Find 
, 
and 
. 
, 
.
A)
, 
, 
B)
, 
, 
C)
, 
, 
D)
, 
, 
E)
, 
, 
, and . , .A)
, , B)
, , C)
, , D)
, , E)
, , Question
Given 
and 
, find 
.
A) 375
B) -250
C) 300
D) 399
E) -210
and , find .A) 375
B) -250
C) 300
D) 399
E) -210
Question
Use the fact that for an 
matrix A, 
, to evaluate 
.
A) -420
B) -1,050
C) -26,250
D) 1,050
E) 26,250
matrix A, , to evaluate .A) -420
B) -1,050
C) -26,250
D) 1,050
E) 26,250
Question
Use the fact that for an 
matrix A, 
, to evaluate 
.
A) -120
B) -360
C) -1,000
D) -40
E) -2,000
matrix A, , to evaluate .A) -120
B) -360
C) -1,000
D) -40
E) -2,000
Question
Evaluate 
for the matrix 
.
A) 40
B) -40
C) -400
D) 400
E) 80
for the matrix .A) 40
B) -40
C) -400
D) 400
E) 80
Question
Evaluate 
for the matrix 
.
A) 7
B) -21
C)
D)
E) -7
for the matrix .A) 7
B) -21
C)
D)
E) -7
Question
Evaluate 
for the matrix 
.
A) 8
B) 784
C) -220
D) 4
E) 484
for the matrix .A) 8
B) 784
C) -220
D) 4
E) 484
Question
Let A and B be square matrices of order 3 such that 
and 
. Find 
. Round your answer to two decimal places when appropriate.
A) 0.02
B) 56
C) 0.88
D) 15
E) 1
and . Find . Round your answer to two decimal places when appropriate.A) 0.02
B) 56
C) 0.88
D) 15
E) 1
Question
Use the determinant to decide whether the matrix given below is singular or nonsingular. 
A) nonsingular
B) singular
A) nonsingular
B) singular
Question
Use the determinant to decide whether the matrix given below is singular or nonsingular. 
A) nonsingular
B) singular
A) nonsingular
B) singular
Question
Find the characteristic equation of the matrix 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
Question
Find the eigenvalues of the matrix 
.
A)
and 
B)
and 
C)
and 
D)
and 
E)
and 
.A)
and B)
and C)
and D)
and E)
and Question
The eigenvalues of the matrix 
are 
and 
. Find an eigenvector corresponding to each of the given eigenvalues.
A)
, 
; 
, 
B)
, 
; 
, 
C)
, 
; 
, 
D)
, 
; 
, 
E)
, 
; 
, 
are and . Find an eigenvector corresponding to each of the given eigenvalues.A)
, ; , B)
, ; , C)
, ; , D)
, ; , E)
, ; , Question
Find the characteristic equation of the matrix 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
Question
Find the eigenvalues of the matrix 
.
A)
and 
B)
and 
C)
and 
D)
and 
E)
and 
.A)
and B)
and C)
and D)
and E)
and Question
The eigenvalues of the matrix 
are 
and 
. Find a non-zero eigenvector associated with each eigenvalue.
A)
, 
; 
, 
B)
, 
; 
, 
C)
, 
; 
, 
D)
, 
; 
, 
E)
, 
; 
, 
are and . Find a non-zero eigenvector associated with each eigenvalue.A)
, ; , B)
, ; , C)
, ; , D)
, ; , E)
, ; , Question
Find the characteristic equation of the matrix 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Find the eigenvalues of the matrix 
. If an eigenvalue has multiplicity greater than one, list the eigenvalue according to its multiplicity. For example, list an eigenvalue with multiplicity two twice.
A)
, 
, and 
B)
, 
, and 
C)
, 
, and 
D)
, 
, and 
E)
, 
, and 
. If an eigenvalue has multiplicity greater than one, list the eigenvalue according to its multiplicity. For example, list an eigenvalue with multiplicity two twice.A)
, , and B)
, , and C)
, , and D)
, , and E)
, , and Question
The eigenvalues of the matrix 
are 
, 
, 
. Find a non-zero eigenvector corresponding to each of the given eigenvalues.
A)
, 
; 
, 
; 
, 
B)
, 
; 
, 
; 
, 
C)
, 
; 
, 
; 
, 
D)
, 
; 
, 
; 
, 
E)
, 
; 
, 
; 
, 
are , , . Find a non-zero eigenvector corresponding to each of the given eigenvalues.A)
, ; , ; , B)
, ; , ; , C)
, ; , ; , D)
, ; , ; , E)
, ; , ; , Question
Find the adjoint of the matrix 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
Question
Use Cramer's Rule to solve the system of equations 
.
A)
, 
B)
, 
C)
, 
D)
, 
E)
, 
.A)
, B)
, C)
, D)
, E)
, Question
Use Cramer's Rule to solve the system of equations 
.
A)x = 5, y = 8
B)x = 5, y = 9
C) x = 4, y = 8
D) x = 7, y = 7
E) x = 9, y = 5
.A)x = 5, y = 8
B)x = 5, y = 9
C) x = 4, y = 8
D) x = 7, y = 7
E) x = 9, y = 5
Question
Use Cramer's Rule to solve the following system of linear equations 
.
A) x =
, y = 
, z = 
B)x =
, y = 
, z = 
C) x =
, y = 
, z = 
D) x =
, y = 
, z = 
E) x =
, y = 
, z = 
.A) x =
, y = , z = B)x =
, y = , z = C) x =
, y = , z = D) x =
, y = , z = E) x =
, y = , z = Question
Find the area of the triangle having the vertices 
, 
, and 
.
A) 12
B) 6
C) 30
D) 69
E) 9
, , and .A) 12
B) 6
C) 30
D) 69
E) 9
Question
Determine whether the points 
, 
, and 
are collinear.
, , and are collinear. Question
Find the volume of the tetrahedron having the vertices 
, 
, 
, and 
.
A) 2
B) 12
C) 26
D) 4
E) 31
, , , and .A) 2
B) 12
C) 26
D) 4
E) 31
Question
Determine whether the points 
, 
, 
, and 
are coplanar.
, , , and are coplanar.
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Deck 3: Determinants
1
Find the determinant of .
A) -14
B) 30
C) 22
D) -6
E) -18
.A) -14
B) 30
C) 22
D) -6
E) -18
22
2
Evaluate the determinant.
A) 16
B) -14
C) 8
D) 2
E) -2
A) 16
B) -14
C) 8
D) 2
E) -2
16
3
Find the determinant of the matrix .
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
4
Find the minor and its cofactor of the matrix .
A)
B)
C)
D)
E)
and its cofactor of the matrix .A)
B)
C)
D)
E)
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5
Find the minor and its cofactor of the matrix .
A)
B)
C)
D)
E)
and its cofactor of the matrix .A)
B)
C)
D)
E)
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6
Find the determinant of by the method of expansion by cofactors.
A) -945
B) -702
C) 594
D) 378
E) 1,350
by the method of expansion by cofactors.A) -945
B) -702
C) 594
D) 378
E) 1,350
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7
Find the determinant of by the method of expansion by cofactors.
A) 192
B) 384
C) -48
D) -384
E) 176
by the method of expansion by cofactors.A) 192
B) 384
C) -48
D) -384
E) 176
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8
Use the matrix capabilities of a graphing utility to find the determinant of the matrix .
A) -0.98
B) -0.343
C) -1.715
D) 1.029
E) -0.686
.A) -0.98
B) -0.343
C) -1.715
D) 1.029
E) -0.686
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9
Use technology to find the determinant.
A) 119
B) -119
C) -207
D) -115
E) 121
A) 119
B) -119
C) -207
D) -115
E) 121
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10
Solve for x given the following equation involving a determinant.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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11
Solve for x given the following equation involving a determinant.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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12
Evaluate the determinant of the matrix below to determine which of the following makes the equation true.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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13
Which property of determinants is illustrated by the equation
A) If every entry of an matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c.
B) If a single row or a single column of an matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c.
C) If a single row or a single column of an matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by .
D) If every entry of an matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by
E) If c times one row is added to another row of an matrix, the determinant of the matrix is multiplied by c.
A) If every entry of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c.B) If a single row or a single column of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c.C) If a single row or a single column of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by .D) If every entry of an
matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by E) If c times one row is added to another row of an
matrix, the determinant of the matrix is multiplied by c. Unlock Deck
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14
Which property of determinants is illustrated by the equation
A) If a multiple of a row is added to another row in a matrix and then the constant is added to a row, the determinant of the matrix is unchanged.
B) If a non-zero constant is added to a row of a matrix, the determinant of the matrix is unchanged.
C) If the two columns of the matrix are interchanged, the determinant of the matrix is unchanged.
D) If a multiple of a row is added to another row of a matrix, the determinant of the matrix is unchanged.
E) If times a row of a matrixis added to another row of a matrix, the determinant of the matrix is multiplied by c.
A) If a multiple of a row is added to another row in a matrix and then the constant is added to a row, the determinant of the matrix is unchanged.
B) If a non-zero constant is added to a row of a matrix, the determinant of the matrix is unchanged.
C) If the two columns of the matrix are interchanged, the determinant of the matrix is unchanged.
D) If a multiple of a row is added to another row of a matrix, the determinant of the matrix is unchanged.
E) If times a row of a matrixis added to another row of a matrix, the determinant of the matrix is multiplied by c.
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15
Which property of determinants is illustrated by the equation
A) If two rows of a matrix are interchanged, the determinant of the matrix is multiplied by -1.
B) If two columns of a matrix are interchanged, the determinant of the matrix is multiplied by -1.
C) If a row of a matrix is subtracted from another row of the same matrix, the determinant of the matrix is multiplied by -1.
D) If two matrices have any rows equal, the determinant of the matrix is multiplied by -1.
E) If a column of a matrix is subtracted from another column, the determinant of the matrix is multiplied by -1.
A) If two rows of a matrix are interchanged, the determinant of the matrix is multiplied by -1.
B) If two columns of a matrix are interchanged, the determinant of the matrix is multiplied by -1.
C) If a row of a matrix is subtracted from another row of the same matrix, the determinant of the matrix is multiplied by -1.
D) If two matrices have any rows equal, the determinant of the matrix is multiplied by -1.
E) If a column of a matrix is subtracted from another column, the determinant of the matrix is multiplied by -1.
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16
Use the matrix capabilities of a graphing utility to find the determinant of the matrix .
A) -162
B) 0
C) -216
D) -188
E) 188
.A) -162
B) 0
C) -216
D) -188
E) 188
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17
Use elementary row or column operations to evaluate the determinant.
A) 160
B) 0
C) -160
D) 120
E) -120
A) 160
B) 0
C) -160
D) 120
E) -120
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18
Use elementary row or column operations to evaluate the determinant.
A) -84
B) 1,812
C) -1,212
D) 180
E) -180
A) -84
B) 1,812
C) -1,212
D) 180
E) -180
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19
Find the determinant of the elementary matrix .
A) -1
B) 1
C) -2
D) 0
E) 2
.A) -1
B) 1
C) -2
D) 0
E) 2
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20
Find the determinant of the elementary matrix .
A) 1
B) 1- k
C) k
D) -k
E) k - 1
.A) 1
B) 1- k
C) k
D) -k
E) k - 1
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21
Find , and . , .
A) , ,
B) , ,
C) , ,
D) , ,
E) , ,
, and . , .A)
, , B)
, , C)
, , D)
, , E)
, , Unlock Deck
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22
Given and , find .
A) 375
B) -250
C) 300
D) 399
E) -210
and , find .A) 375
B) -250
C) 300
D) 399
E) -210
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23
Use the fact that for an matrix A, , to evaluate .
A) -420
B) -1,050
C) -26,250
D) 1,050
E) 26,250
matrix A, , to evaluate .A) -420
B) -1,050
C) -26,250
D) 1,050
E) 26,250
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24
Use the fact that for an matrix A, , to evaluate .
A) -120
B) -360
C) -1,000
D) -40
E) -2,000
matrix A, , to evaluate .A) -120
B) -360
C) -1,000
D) -40
E) -2,000
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25
Evaluate for the matrix .
A) 40
B) -40
C) -400
D) 400
E) 80
for the matrix .A) 40
B) -40
C) -400
D) 400
E) 80
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26
Evaluate for the matrix .
A) 7
B) -21
C)
D)
E) -7
for the matrix .A) 7
B) -21
C)
D)
E) -7
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27
Evaluate for the matrix .
A) 8
B) 784
C) -220
D) 4
E) 484
for the matrix .A) 8
B) 784
C) -220
D) 4
E) 484
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28
Let A and B be square matrices of order 3 such that and . Find . Round your answer to two decimal places when appropriate.
A) 0.02
B) 56
C) 0.88
D) 15
E) 1
and . Find . Round your answer to two decimal places when appropriate.A) 0.02
B) 56
C) 0.88
D) 15
E) 1
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29
Use the determinant to decide whether the matrix given below is singular or nonsingular.
A) nonsingular
B) singular
A) nonsingular
B) singular
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30
Use the determinant to decide whether the matrix given below is singular or nonsingular.
A) nonsingular
B) singular
A) nonsingular
B) singular
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31
Find the characteristic equation of the matrix .
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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32
Find the eigenvalues of the matrix .
A) and
B) and
C) and
D) and
E) and
.A)
and B)
and C)
and D)
and E)
and Unlock Deck
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33
The eigenvalues of the matrix are and . Find an eigenvector corresponding to each of the given eigenvalues.
A) , ; ,
B) , ; ,
C) , ; ,
D) , ; ,
E) , ; ,
are and . Find an eigenvector corresponding to each of the given eigenvalues.A)
, ; , B)
, ; , C)
, ; , D)
, ; , E)
, ; , Unlock Deck
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34
Find the characteristic equation of the matrix .
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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35
Find the eigenvalues of the matrix .
A) and
B) and
C) and
D) and
E) and
.A)
and B)
and C)
and D)
and E)
and Unlock Deck
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36
The eigenvalues of the matrix are and . Find a non-zero eigenvector associated with each eigenvalue.
A) , ; ,
B) , ; ,
C) , ; ,
D) , ; ,
E) , ; ,
are and . Find a non-zero eigenvector associated with each eigenvalue.A)
, ; , B)
, ; , C)
, ; , D)
, ; , E)
, ; , Unlock Deck
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37
Find the characteristic equation of the matrix .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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38
Find the eigenvalues of the matrix . If an eigenvalue has multiplicity greater than one, list the eigenvalue according to its multiplicity. For example, list an eigenvalue with multiplicity two twice.
A) , , and
B) , , and
C) , , and
D) , , and
E) , , and
. If an eigenvalue has multiplicity greater than one, list the eigenvalue according to its multiplicity. For example, list an eigenvalue with multiplicity two twice.A)
, , and B)
, , and C)
, , and D)
, , and E)
, , and Unlock Deck
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39
The eigenvalues of the matrix are , , . Find a non-zero eigenvector corresponding to each of the given eigenvalues.
A) , ; , ; ,
B) , ; , ; ,
C) , ; , ; ,
D) , ; , ; ,
E) , ; , ; ,
are , , . Find a non-zero eigenvector corresponding to each of the given eigenvalues.A)
, ; , ; , B)
, ; , ; , C)
, ; , ; , D)
, ; , ; , E)
, ; , ; , Unlock Deck
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40
Find the adjoint of the matrix .
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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41
Use Cramer's Rule to solve the system of equations .
A) ,
B) ,
C) ,
D) ,
E) ,
.A)
, B)
, C)
, D)
, E)
, Unlock Deck
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42
Use Cramer's Rule to solve the system of equations .
A)x = 5, y = 8
B)x = 5, y = 9
C) x = 4, y = 8
D) x = 7, y = 7
E) x = 9, y = 5
.A)x = 5, y = 8
B)x = 5, y = 9
C) x = 4, y = 8
D) x = 7, y = 7
E) x = 9, y = 5
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43
Use Cramer's Rule to solve the following system of linear equations .
A) x = , y = , z =
B)x = , y = , z =
C) x = , y = , z =
D) x = , y = , z =
E) x = , y = , z =
.A) x =
, y = , z = B)x =
, y = , z = C) x =
, y = , z = D) x =
, y = , z = E) x =
, y = , z = Unlock Deck
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44
Find the area of the triangle having the vertices , , and .
A) 12
B) 6
C) 30
D) 69
E) 9
, , and .A) 12
B) 6
C) 30
D) 69
E) 9
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45
Determine whether the points , , and are collinear.
, , and are collinear. Unlock Deck
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46
Find the volume of the tetrahedron having the vertices , , , and .
A) 2
B) 12
C) 26
D) 4
E) 31
, , , and .A) 2
B) 12
C) 26
D) 4
E) 31
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47
Determine whether the points , , , and are coplanar.
, , , and are coplanar. Unlock Deck
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