Chapter 5 Determinants- test bank-Introduction to Linear Algebra for Scientists & Engineers

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Introduction to Linear Algebra for Science and Engineering, 2e (Norman & Wolczuk)
Chapter 5 Determinants

5.1 Determinants in Terms of Cofactors

Evaluate the determinant by expanding along the first row

Evaluate the determinant by expanding along the first row.
1)
A) 3
B) 4
C) 2
D) 1
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Recall
Objective: Evaluate a determinant by expanding along the first row

2)
A) 1
B) -1
C) 0
D) 3
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Recall
Objective: Evaluate a determinant by expanding along the first row

Demonstrate knowledge of determinant theorems

Find the determinant of the matrix.
1)
A) 200
B) -100
C) 0
D) -200
Answer: D
Diff: 2 Type: BI Var: 50+
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Recall
Objective: Demonstrate knowledge of determinant theorems
Evaluate a determinant by using a cofactor expansion along any row or column

Evaluate the determinant by using a cofactor expansion along any row or column.
1)
A) 150
B) -150
C) -30
D) 0
Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Recall
Objective: Evaluate a determinant by using a cofactor expansion along any row or column

2)
A) 64
B) -64
C) -78
D) 50
Answer: B
Diff: 1 Type: BI Var: 50+
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Recall
Objective: Evaluate a determinant by using a cofactor expansion along any row or column

3)
A) -48
B) 48
C) -57
D) 39
Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Recall
Objective: Evaluate a determinant by using a cofactor expansion along any row or column
4)
A) -38
B) 24
C) 38
D) 14
Answer: C
Diff: 1 Type: BI Var: 50+
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Recall
Objective: Evaluate a determinant by using a cofactor expansion along any row or column

5)
A) -11
B) 213
C) 11
D) -35
Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Recall
Objective: Evaluate a determinant by using a cofactor expansion along any row or column

6)
A) 39
B) 955
C) -39
D) -459
Answer: C
Diff: 1 Type: BI Var: 50+
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Recall
Objective: Evaluate a determinant by using a cofactor expansion along any row or column

Determine the determinant of an elementary matrix

1)
A) -1
B) 1
C) 0
D) 2
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Applied
Objective: Determine the determinant of an elementary matrix

2)
A) -k
B) 0
C) k
D) 1
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Applied
Objective: Determine the determinant of an elementary matrix

3)
A) 4
B) 0
C) -4
D) 1
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Applied
Objective: Determine the determinant of an elementary matrix

4)
A) -1
B) -k
C) k
D) 1
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Applied
Objective: Determine the determinant of an elementary matrix

5)
A) 1
B) -5
C) 5
D) -1
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Applied
Objective: Determine the determinant of an elementary matrix

6)
A) -1
B) -k
C) k
D) 1
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Applied
Objective: Determine the determinant of an elementary matrix

7)
A) 1
B) -k
C) k
D) -1
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Applied
Objective: Determine the determinant of an elementary matrix

8)
A) 1
B) -1
C) 0
D) 2
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Applied
Objective: Determine the determinant of an elementary matrix

9)
A) 1
B) -k
C) k
D) -1
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (5.1) Determinants in Terms of Cofactors
Skill: Applied
Objective: Determine the determinant of an elementary matrix

5.2 Elementary Row Operations and the Determinant

Evaluate a determinant by row reducing to upper triangular form

1)
A) -36
B) 36
C) 0
D) -12
Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Evaluate a determinant by row reducing to upper triangular form

2)
A) 30
B) 15
C) -30
D) 0
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Evaluate a determinant by row reducing to upper triangular form

3)
A) 56
B) 14
C) 28
D) -28
Answer: A
Diff: 2 Type: BI Var: 4
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Evaluate a determinant by row reducing to upper triangular form

4)
A) 80
B) -80
C) -160
D) -40
Answer: C
Diff: 2 Type: BI Var: 18
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Evaluate a determinant by row reducing to upper triangular form

5)
A) 5
B) -10
C) -5
D) 10
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Evaluate a determinant by row reducing to upper triangular form

Use a combination of row operations and cofactor expansions to evaluate a determinant

1)
A) -160
B) -80
C) 80
D) -40
Answer: A
Diff: 2 Type: BI Var: 18
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Use a combination of row operations and cofactor expansions to evaluate a determinant

2)
A) -1
B) 1
C) 0
D) 80
Answer: C
Diff: 2 Type: BI Var: 50+
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Use a combination of row operations and cofactor expansions to evaluate a determinant

3) Let A = . Then det(A) is ________.
A) 0
B) a + b + c
C) (a – b)(b – c)(c – a)
D) (a – b)(b -c )(a – c)
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Use a combination of row operations and cofactor expansions to evaluate a determinant

Determine if a matrix is invertible by checking if the determinant is non-zero

1) Determine if the matrix is invertible. If so, find its inverse.
A) It is invertible and the inverse is .

B) It is invertible and the inverse is .

C) It is invertible and the inverse is .

D) It is not invertible.

Answer: D
Diff: 2 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero
Determine if a matrix is invertible by checking if the determinant is non-zero.
2) For what values of p is the matrix not invertible?
A) –
B)
C) –
D)
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero

3) Determine if the matrix is invertible. If so, find the inverse.
A) It is invertible and the inverse is .

B) It is invertible and the inverse is .

C) It is invertible and the inverse is .

D) It is not invertible.

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero
Determine if a matrix is invertible by checking if the determinant is non-zero.
4) For what values of p is the matrix not invertible?
A)
B) 32
C) 16
D) 8
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero

5) If A is an n × n invertible matrix, then find det(ATA-1).
A) 1
B) 0
C) -1
D) undefined
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero

6) Suppose that det = 2, and A = . Find det(2A-1).
A) 12
B) 1
C) 3
D) 4
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero

7) Let A and B be n × n invertible matrices, then det(B-1 AB) = ________.
A) det(B)
B) 0
C) det(A)
D) -det(A)
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero
8) Let A and B be 3 × 3 matrices with det(A) = 2 and det(B) = -3. Then find det(-A2B).
A) 12
B) -12
C) 6
D) -6
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero

9) Suppose that A is a 4 × 4 matrix and det(A) = -3. If C is obtained from A by switching the second column and the first column, what is det(C)?
A) -3
B) 3
C) 0
D) 12
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Recall
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero

10) Suppose that A is a 4 × 4 matrix and det(A) = -3. If B is obtained from A by multiplying the second column of A by , what is det(B)?
A)
B) –
C) 6
D) 3
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Recall
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero

11) Suppose that A is a 4 × 4 matrix and det(A) = -1. What is det(3AAT)?
A) 9
B) 3
C) -3
D) 81
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Recall
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero
12) Let A and B be 3 × 3 matrix with det(A) = 2 and det(B) = -3. Then find det(-A2B).
A) 12
B) -12
C) 6
D) -6
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (5.2) Elementary Row Operations and the Determinant
Skill: Applied
Objective: Determine if a matrix is invertible by checking if the determinant is non-zero

5.3 Matrix Inverse by Cofactors and Cramer’s Rule

Find the inverse of a matrix by cofactors

1) Suppose that A-1 = . Then find C of A.
A)

B)

C)

D)

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Find the inverse of a matrix by cofactors

2) Find det(C of (A)) if A = .
A) -192
B) 384
C) 192
D) -3843
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Find the inverse of a matrix by cofactors

Find the inverse of a matrix by cofactors.
3) Let A = . Then find the inverse of A.

A)

B)

C)

D)

Answer: B
Diff: 2 Type: BI Var: 1
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Find the inverse of a matrix by cofactors

4) Let A = . Then find the inverse of A.

A)

B)

C)

D)

Answer: B
Diff: 2 Type: BI Var: 1
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Find the inverse of a matrix by cofactors

5)

A)

B)

C)

D)

Answer: A
Diff: 3 Type: BI Var: 1
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Find the inverse of a matrix by cofactors

6)

A)

B)

C)

D)

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Find the inverse of a matrix by cofactors

7) Let A = . Determine (A-1)13 by using the cofactor method.
A)
B)
C) –
D) –
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Find the inverse of a matrix by cofactors

8) Suppose that A is an invertible 3 × 3 matrix with det (A) = 3. Then find det(A-1 + (Cof A)T).
A)
B)
C)
D) undefined
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Find the inverse of a matrix by cofactors

9) Assume A is an invertible 3 × 3 matrix. Compute det(-A-2(C of A)).
A) 1
B) 2
C) -1
D) 0
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Find the inverse of a matrix by cofactors

Solve a system of equations using Cramer’s Rule

Solve a system of equations using Cramer’s Rule.
1) 2×1 + 6×2 = 32
4×1 + x2 = -2
A) (-2, -6)
B) (6, -2)
C) (-6, -2)
D) (-2, 6)
Answer: D
Diff: 1 Type: BI Var: 50+
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Solve a system of equations using Cramer’s Rule

2) 4×1 + 2×2 = 34
-2×1 + 3×2 = -5
A) (-7, -3)
B) (3, 7)
C) (-3, 7)
D) (7, 3)
Answer: D
Diff: 1 Type: BI Var: 50+
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Solve a system of equations using Cramer’s Rule
3) 2×1 + 3×2 = 28
3×1 – 2×2 = 3
A) (-5, 6)
B) (6, 5)
C) (-6, 5)
D) (5, 6)
Answer: D
Diff: 1 Type: BI Var: 50+
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Solve a system of equations using Cramer’s Rule

4) 2sx1 + 4×2 = -3
2×1 + sx2 = 4
A) s ≠ ± 2; x1 = and x2 =
B) s ≠ ± 2; x1 = and x2 =
C) s ≠ 2; x1 = and x2 =
D) s ≠ ± 4; x1 = -3s – 16 and x2 = 4s + 3
Answer: A
Diff: 3 Type: BI Var: 4
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Solve a system of equations using Cramer’s Rule

5) sx1 – 7sx2 = 3
3×1 – 21sx2 = 5
A) s ≠ 0, 1; x1 = and x2 =
B) s ≠ ± 1; x1 = and x2 =
C) s ≠ 1; x1 = and x2 =
D) s ≠ 1; x1 = and x2 =
Answer: A
Diff: 3 Type: BI Var: 15
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Solve a system of equations using Cramer’s Rule
6) Solve:
2x + y + z = a
x + 2y + z = b
x + y + 2z = c
A) x = , y = , z =
B) x = , y = , z =
C) x = , y = , z =
D) x = , y = , z =
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (5.3) Matrix Inverse by Cofactors and Cramer’s Rule
Skill: Applied
Objective: Solve a system of equations using Cramer’s Rule

5.4 Area, Volume, and the Determinant

Calculate the area of a parallelogram using the determinant

Calculate the area of the given parallelogram using the determinant.
1) Calculate the area of the parallelogram induced by = , = .
A) -1
B) 1
C) 7
D) -3
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (5.4) Area, Volume, and the Determinant
Skill: Applied
Objective: Calculate the area of a parallelogram using the determinant

2) (0, 0), (5, 8), (13, 10), (8, 2)
A) 54
B) 108
C) 59
D) 53
Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (5.4) Area, Volume, and the Determinant
Skill: Applied
Objective: Calculate the area of a parallelogram using the determinant
3) (-1, -2), (3, 6), (5, 0), (9, 8)
A) 44
B) 80
C) 40
D) 39
Answer: C
Diff: 2 Type: BI Var: 50+
Topic: (5.4) Area, Volume, and the Determinant
Skill: Applied
Objective: Calculate the area of a parallelogram using the determinant

Compute the area of the image parallelogram under a linear mapping

1) Let A = , = , = , = . Calculate the volume of the parallelepiped induced by A , A , and A .
A) 12
B) 105
C) 109
D) 153
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (5.4) Area, Volume, and the Determinant
Skill: Applied
Objective: Compute the area of the image parallelogram under a linear mapping

2) Let A = be the standard matrix of the stretch S: R2 → R2 in the x direction by a factor of t. Compute the area determined by the image vectors of and .
A) 2t
B) t
C) 0
D) 2
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (5.4) Area, Volume, and the Determinant
Skill: Applied
Objective: Compute the area of the image parallelogram under a linear mapping
Compute the volume of a parallepiped using the determinant

1) Calculate the volume of the 4-dimensional parallelotope determined by , , , and .
A) 2
B) 1
C) 3
D) -2
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (5.4) Area, Volume, and the Determinant
Skill: Applied
Objective: Compute the volume of a parallepiped using the determinant

Compute the volume of a parallepiped using the determinant.
2) Compute the volume of the parallepiped determined by , , and .
A) 3
B) -7
C) 7
D) 5
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (5.4) Area, Volume, and the Determinant
Skill: Applied
Objective: Compute the volume of a parallepiped using the determinant

3) Let A = , = , = , and = . Calculate the volume of the parallepiped induced by
A , A and A .
A) 3
B) 4
C) -7
D) 7
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (5.4) Area, Volume, and the Determinant
Skill: Applied
Objective: Compute the volume of a parallepiped using the determinant