Chapter 3 Matrices- Linear Mappings and Inverses- test bank-Introduction to Linear Algebra for Scientists & Engineers

Download file with the answers

Chapter 3 Matrices- Linear Mappings and Inverses- test bank-Introduction to Linear Algebra for Scientists & Engineers
1 file(s) 2.85 MB

Not a member!
Create a FREE account here to get access and download this file with answers

Introduction to Linear Algebra for Science and Engineering, 2e (Norman & Wolczuk)
Chapter 3 Matrices, Linear Mappings, and Inverses

3.1 Operations on Matrices

Calculate linear combinations of matrices

Perform the matrix operation.
1) Let A = . Find 2A.
A)

Answer: A
Diff: 1 Type: BI Var: 24
Topic: (3.1) Operations on Matrices
Skill: Recall
Objective: Calculate linear combinations of matrices

2) Let B = . Find -4B.
A)
B)
C)
D)
Answer: C
Diff: 1 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Recall
Objective: Calculate linear combinations of matrices

3) Let C = . Find (1/2)C.

Answer: B
Diff: 1 Type: BI Var: 9
Topic: (3.1) Operations on Matrices
Skill: Recall
Objective: Calculate linear combinations of matrices

4) Let A = and B = . Find 4A + B.

Answer: A
Diff: 1 Type: BI Var: 27
Topic: (3.1) Operations on Matrices
Skill: Recall
Objective: Calculate linear combinations of matrices

5) Let C = and D = . Find C – 4D.

Answer: C
Diff: 1 Type: BI Var: 3
Topic: (3.1) Operations on Matrices
Skill: Recall
Objective: Calculate linear combinations of matrices

6) Let A = and B = . Find 3A + 4B.
A)
B)
C)
D)
Answer: A
Diff: 1 Type: BI Var: 10
Topic: (3.1) Operations on Matrices
Skill: Recall
Objective: Calculate linear combinations of matrices

7) Let A = and B = . Find A + B.

Answer: D
Diff: 1 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Recall
Objective: Calculate linear combinations of matrices

8) Let A = and B = . Find A – B.

Answer: C
Diff: 1 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Recall
Objective: Calculate linear combinations of matrices

9) Let A = and B = . Find A + B.

D) Undefined

Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Recall
Objective: Calculate linear combinations of matrices

Calculate matrix-vector and matrix-matrix products possibly with tranposes

Find the transpose of the matrix.
1)

Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Recall
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

Answer: B
Diff: 1 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Recall
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

Find the matrix product AB, if it is defined.
3) A = , B =

Answer: C
Diff: 2 Type: BI Var: 30
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

4) A = , B =

Answer: A
Diff: 2 Type: BI Var: 36
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

5) A = , B =

Answer: B
Diff: 2 Type: BI Var: 24
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

6) A = , B =

B) AB is undefined.

Answer: C
Diff: 2 Type: BI Var: 36
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

7) A = , B =
A) AB is undefined.

Answer: A
Diff: 2 Type: BI Var: 27
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

8) A = , B =

B) AB is undefined.

Answer: C
Diff: 2 Type: BI Var: 27
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

9) A = , B =

B) AB is undefined.

Answer: A
Diff: 2 Type: BI Var: 27
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

10) A = , B =

B) AB is undefined.

Answer: D
Diff: 2 Type: BI Var: 24
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

Compute the product or state that it is undefined.
11) [-6 2 5]

A) [-51]

B) [174]

C) Undefined

Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

12)

A) Undefined

C) [-63 -30]

Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

13)

D) Undefined

Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

14)

A) Undefined

Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

Write the system as a vector equation or matrix equation as indicated.
15) Write the following system as a matrix equation involving the product of a matrix and a vector on the left side and a vector on the right side.
4×1 + x2 – 2×3 = 5
2×1 – 3×2 = 1
A) =
B) =
C) =
D) =
Answer: D
Diff: 2 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined.
16) A is 4 × 4, B is 4 × 4.
A) AB is 4 × 4, BA is 4 × 4.
B) AB is 1 × 1, BA is 1 × 1.
C) AB is 4 × 8, BA is 4 × 8.
D) AB is 8 × 4, BA is 8 × 4.
Answer: A
Diff: 2 Type: BI Var: 3
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

17) A is 2 × 1, B is 1 × 1.
A) AB is 2 × 2, BA is 1 × 1.
B) AB is 2 × 1, BA is undefined.
C) AB is 1 × 2, BA is 1 × 1.
D) AB is undefined, BA is 1× 2.
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

18) A is 2 × 3, B is 3 × 2.
A) AB is 2 × 2, BA is undefined.
B) AB is undefined, BA is 3 × 3.
C) AB is 2 × 2, BA is 3 × 3.
D) AB is 3 × 3, BA is 2 × 2.
Answer: C
Diff: 2 Type: BI Var: 12
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

19) A is 2 × 1, B is 2 × 1.
A) AB is undefined, BA is undefined.
B) AB is 1 × 2, BA is 2 × 1.
C) AB is 2 × 1, BA is 1 × 2.
D) AB is 2 × 2, BA is 1 × 1.
Answer: A
Diff: 2 Type: BI Var: 12
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate matrix-vector and matrix-matrix products possibly with tranposes

Calculate a matrix product using block matrices

Identify the indicated sub matrix.
1) A = . Find A12.

A)
B) 1
C)
D)
Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate a matrix product using block matrices

2) A = . Find A21.
A)
B)
C)
D)
Answer: C
Diff: 2 Type: BI Var: 50+
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate a matrix product using block matrices

Find the matrix product AB for the partitioned matrices.
3) A = , B =

Answer: A
Diff: 3 Type: BI Var: 1
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate a matrix product using block matrices

4) A = , B =

Answer: D
Diff: 3 Type: BI Var: 1
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Calculate a matrix product using block matrices

Determine if a matrix is in the span of a set of matrices

Determine if a matrix is in the span of a set of matrices.
1) Consider Β = , A = , and C = .
A) Neither A nor B is in the Span B.
B) Only A is in the Span Β .
C) Both A and C are in the Span B.
D) Only C is in the Span Β.
Answer: B
Diff: 3 Type: MC Var: 1
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Determine if a matrix is in the span of a set of matrices

Determine if a set of matrices is linearly independent

Determine which of the sets of matrices is linearly independent.
1) A = , B =
A) Neither A nor B
B) A only
C) Both A and B
D) B only
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (3.1) Operations on Matrices
Skill: Applied
Objective: Determine if a set of matrices is linearly independent

3.2 Matrix Mappings and Linear Mappings

Find the domain and codomain of a linear mapping

1) Let A = and fA be the corresponding matrix mapping.
Determine the domain and codomain of fA.
A) Domain = ℛ3, Codomain = ℛ3.
B) Domain = ℛ2, Codomain = ℛ2.
C) Domain = ℛ3, Codomain = ℛ2.
D) Domain = ℛ2, Codomain = ℛ3.
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Recall
Objective: Find the domain and codomain of a linear mapping

Evaluate L(x) using a formula or the standard matrix

Find the standard matrix of the following linear mapping.
1) Let ℒ: ℛ3 → ℛ3 be a linear mapping defined by ℒ(x, y, z) = (5x, 4y, z).
A)

Answer: C
Diff: 2 Type: BI Var: 50+
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Evaluate L(x) using a formula or the standard matrix

Solve the problem.
2) Let A = and u = .
Define a transformation T: ℛ3 → ℛ2 by T(x) = Ax. Find T(u), the image of u under the transformation T.
A)

Answer: C
Diff: 2 Type: BI Var: 50+
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Evaluate L(x) using a formula or the standard matrix

3) Let T: ℛ2 → ℛ2 be a linear transformation that maps u = into and maps v = into .
Use the fact that T is linear to find the image of
A)

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Evaluate L(x) using a formula or the standard matrix

4) The columns of I3 = are e1 = , e2 = , and e3 = .

Suppose that T is a linear transformation from ℛ3 into ℛ2 such that

T(e1) = , T(e2) = , and T(e3) = .

Find a formula for the image of an arbitrary x = in ℛ3.
A) T =

B) T =

C) T =

D) T =

Answer: D
Diff: 3 Type: BI Var: 50+
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Find the standard matrix of a linear mapping

Determine if a mapping is linear

1) Determine which of the following mappings are linear.
f(x1, x2) = (x1 + 2, x2) and g(x1, x2) = (|x1|, x2).
A) g only
B) f only
C) Neither
D) Both
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Determine if a mapping is linear

Find the standard matrix of a linear mapping

Find the matrix of the linear transformation T: V → W relative to B and C.
1) Suppose B = {b1, b2} is a basis for V, and C = {c1, c2, c3} is a basis for W. Let T be defined by
T(b1) = -5c1 – 6c2 + 5c3
T(b2) = -5c1 – 12c2 + 2c3
A)

Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Find the standard matrix of a linear mapping

2) Suppose B = {b1, b2, b3} is a basis for V, and C = {c1, c2} is a basis for W. Let T be defined by
T(b1) = 5c1 + c2
T(b2) = 6c1 – 6c2
T(b3) = 5c1 – 6c2
A)

Answer: D
Diff: 2 Type: BI Var: 50+
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Find the standard matrix of a linear mapping

Find the standard matrix of the following linear mapping.
3) Reflect through the x-axis R3.
A)

Answer: A
Diff: 2 Type: BI Var: 1
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Evaluate L(x) using a formula or the standard matrix

Find the 4 x 4 standard matrix of the linear mapping.
4) Translation by the vector (4, -7, -9).
A)

Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Evaluate L(x) using a formula or the standard matrix

Determine the standard matrix of a composition of mappings

1) Suppose that S and T are linear mappings with matrices [S] = and {T] = .
Determine the matrix that represents S ∘T, if it is defined.

D) undefined

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Determine the standard matrix of a composition of mappings

2) Suppose that S and T are linear mappings with matrices [S] = and {T] = .
Determine the matrix that represents T ∘S, if it is defined.

D) undefined

Answer: D
Diff: 3 Type: BI Var: 1
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Determine the standard matrix of a composition of mappings

Determine the standard matrix of a projection

1) Let = . Determine the matrix .
A)

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (3.2) Matrix Mappings and Linear Mappings
Skill: Applied
Objective: Determine the standard matrix of a projection

3.3 Geometrical Transformations

Determine the standard matrix of a rotation and use it to calculate a rotation

Find the standard matrix of the linear transformation T.
1) T: ℛ2 → ℛ2 rotates points (about the origin) through π radians (with counterclockwise rotation for a positive angle).

Answer: A
Diff: 3 Type: BI Var: 2
Topic: (3.3) Geometrical Transformations
Skill: Applied
Objective: Determine the standard matrix of a rotation and use it to calculate a rotation

Determine the standard matrix of a composition of geometric mappings including projection, reflection

Find the 4 x 4 standard matrix of the linear mapping.
1) Rotation about the y-axis through an angle of 60°
A)

Answer: A
Diff: 3 Type: BI Var: 1
Topic: (3.3) Geometrical Transformations
Skill: Applied
Objective: Determine the standard matrix of a composition of geometric mappings including projection, reflection

Describe geometrically the effect of the transformation T.
2) Let A = .
Define a transformation T by T(x) = Ax.
A) Rotation
B) Projection onto the x2-axis
C) Projection onto the x2x3-plane
D) Reflection
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (3.3) Geometrical Transformations
Skill: Applied
Objective: Determine the standard matrix of a composition of geometric mappings including projection, reflection

Determine the standard matrix of the composition of mappings.
3) Rotate points through 45° and then scale the x-coordinate by 0.2 and the y-coordinate by 0.4.

Answer: A
Diff: 3 Type: BI Var: 12
Topic: (3.3) Geometrical Transformations
Skill: Applied
Objective: Determine the standard matrix of a composition of geometric mappings including projection, reflection

Determine the 3×3 standard matrix of the composition of mappings.
4) Translate by x coordinate by 9 and y coordinate by 7 and then reflect through the line y = x.

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (3.3) Geometrical Transformations
Skill: Applied
Objective: Determine the standard matrix of a composition of geometric mappings including projection, reflection

3.4 Special Subspaces for Systems and Mappings: Rank Theorem

Determine if a vector in the range of a linear mapping

Solve the problem.
1) Let A = and b = .
Define a transformation T: ℛ3 → ℛ3 by T(x) = Ax.
If possible, find a vector x whose image under T is b. Otherwise, state that b is not in the range of the transformation T.
A)

D) b is not in the range of the transformation T.

Answer: B
Diff: 3 Type: BI Var: 50+
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Determine if a vector in the range of a linear mapping

2) Let A = and b = .
Define a transformation T: ℛ3 → ℛ3 by T(x) = Ax.
If possible, find a vector x whose image under T is b. Otherwise, state that b is not in the range of the transformation T.
A) b is not in the range of the transformation T.

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Evaluate L(x) using a formula or the standard matrix

Determine if the vector u is in the column space of matrix A and whether it is in the null space of A.
3) u = , A =

A) Not in Col A, in Nul A
B) In Col A and in Nul A
C) Not in Col A, not in Nul A
D) In Col A, not in Nul A
Answer: C
Diff: 3 Type: BI Var: 50+
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Determine if a vector in the range of a linear mapping

4) Determine which of the following vectors belongs to the null space of the matrix A.
A = , = , and = .

A) None of them
B) only
C) and
D) only
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Determine if a vector in the range of a linear mapping

Find a basis for the range and kernel of a linear mapping

Find a basis for the kernel of the linear mapping fA.
1) A =

Answer: B
Diff: 3 Type: BI Var: 36
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for the range and kernel of a linear mapping

2) A =

Answer: C
Diff: 3 Type: BI Var: 50+
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for the range and kernel of a linear mapping

3) A =

A) ,

B) , ,

C) ,

D) , ,

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for the range and kernel of a linear mapping

4) A =

A) ,

B) , ,

C) , , ,

D) , ,

Answer: D
Diff: 3 Type: BI Var: 10
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for the range and kernel of a linear mapping

Find a basis for the column space of the matrix.
5) B =

Answer: A
Diff: 3 Type: BI Var: 47
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for the range and kernel of a linear mapping

6) B =

Answer: B
Diff: 3 Type: BI Var: 50+
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for the range and kernel of a linear mapping

Determine the rank of a matrix

Assume that the matrix A is row equivalent to B. Find a basis for the row space of the matrix A.
1) A = , B =

A) {(1, 3, -4, 0, 1), (0, -2, 3, 5, -4), (0, 0, -8, -23, 17)}
B) {(1, 0, 0, 0), (3, -2, 0, 0), (-4, 3, -8, 0)}
C) {(1, 3, -4, 0, 1), (2, 4, -5, 5, -2), (1, -5, 0, -3, 2), (-3, -1, 8, 3, -4)}
D) {(1, 3, -4, 0, 1), (0, -2, 3, 5, -4), (0, 0, -8, -23, 17), (0, 0, 0, 0, 0)}
Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Determine the rank of a matrix
Solve the problem.
2) If the null space of a 7 × 5 matrix is 2-dimensional, find Rank A, Dim Row A, and Dim Col A.
A) Rank A = 5, Dim Row A = 5, Dim Col A = 3
B) Rank A = 3, Dim Row A = 3, Dim Col A = 2
C) Rank A = 3, Dim Row A = 3, Dim Col A = 3
D) Rank A = 3, Dim Row A = 2, Dim Col A = 2
Answer: C
Diff: 3 Type: BI Var: 49
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Determine the rank of a matrix

3) If A is a 5 × 9 matrix, what is the smallest possible dimension of Nul A?
A) 0
B) 5
C) 9
D) 4
Answer: D
Diff: 3 Type: BI Var: 10
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Determine the rank of a matrix

4) Find the rank of the matrix A = .
A) 4
B) 2
C) 3
D) 5
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Determine the rank of a matrix
Find a basis for Row(A), Col(A), and Null(A) and verify the Rank Theorem

Determine whether the linear transformation T is one-to-one and whether it maps as specified.
1) Let T be the linear transformation whose standard matrix is
A = .
Determine whether the linear transformation T is one-to-one and whether it maps ℛ3 onto ℛ3.
A) One-to-one; onto ℛ3
B) One-to-one; not onto ℛ3
C) Not one-to-one; onto ℛ3
D) Not one-to-one; not onto ℛ3
Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for Row(A), Col(A), and Null(A) and verify the Rank Theorem

2) Let A = .
Find a basis for the row space, a subset for a basis for the column space and a basis for the null space. Verify the rank theorem.
A) A basis for the row space is .
A basis for the column space is .
A basis for the null space is .

B) A basis for the row space is .
A basis for the column space is .
A basis for the null space is .

C) A basis for the row space is .
A basis for the column space is .
A basis for the null space is .

D) A basis for the row space is .
A basis for the column space is .
A basis for the null space is .

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for Row(A), Col(A), and Null(A) and verify the Rank Theorem

Determine whether the linear transformation T is one-to-one and whether it maps as specified.
3) T(x1, x2, x3) = (-2×2 – 2×3, -2×1 + 8×2 + 4×3, -x1 – 2×3, 4×2 + 4×3)
Determine whether the linear transformation T is one-to-one and whether it maps ℛ3 onto ℛ4.
A) Not one-to-one; onto ℛ4
B) One-to-one; not onto ℛ4
C) Not one-to-one; not onto ℛ4
D) One-to-one; onto ℛ4
Answer: C
Diff: 3 Type: BI Var: 36
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for Row(A), Col(A), and Null(A) and verify the Rank Theorem

Find the dimensions of the null space and the column space of the given matrix.
4) A =
A) dim Nul A = 2, dim Col A = 3
B) dim Nul A = 3, dim Col A = 2
C) dim Nul A = 3, dim Col A = 3
D) dim Nul A = 4, dim Col A = 1
Answer: B
Diff: 3 Type: BI Var: 50+
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for Row(A), Col(A), and Null(A) and verify the Rank Theorem
5) A =
A) dim Nul A = 3, dim Col A = 4
B) dim Nul A = 4, dim Col A = 3
C) dim Nul A = 2, dim Col A = 5
D) dim Nul A = 5, dim Col A = 2
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for Row(A), Col(A), and Null(A) and verify the Rank Theorem

Find a basis for the column space of the matrix.
6) Find a basis for Col B where

B = .
A) , , ,

B) , , , , ,

C) , ,

D) , , ,
Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for Row(A), Col(A), and Null(A) and verify the Rank Theorem

7) Let A = and B = .

It can be shown that matrix A is row equivalent to matrix B. Find a basis for Col A.
A) , ,

B) , ,

C) , , , ,

D) , ,
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (3.4) Special Subspaces for Systems and Mappings: Rank Theorem
Skill: Applied
Objective: Find a basis for Row(A), Col(A), and Null(A) and verify the Rank Theorem

3.5 Inverse Matrices and Inverse Mappings

Find the inverse of a square matrix or show that it is not invertible

Find the inverse of the matrix, if it exists.
1) A =

Answer: C
Diff: 2 Type: BI Var: 50+
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

2) A =

Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

3) A =

D) A is not invertible.

Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

4) A =

A) A is not invertible.

Answer: A
Diff: 2 Type: BI Var: 10
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

5) A =

Answer: D
Diff: 2 Type: BI Var: 50+
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

6) A =

Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

Answer: A
Diff: 3 Type: BI Var: 1
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

Find the inverse of the matrix A, if it exists.
8) A =

A) =

B) =

C) =

D) does not exist.

Answer: D
Diff: 2 Type: BI Var: 50+
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

9) A =

A) =

B) =

C) =

D) does not exist.

Answer: A
Diff: 2 Type: BI Var: 1
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

10) A =

A) does not exist.

B) =

C) =

D) =

Answer: D
Diff: 2 Type: BI Var: 1
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

11) A =

A) =

B) =

C) =

D) does not exist.

Answer: C
Diff: 2 Type: BI Var: 1
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

12) A =

A) does not exist.

B) =

C) =

D) =

Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

13) A =

A) =

B) =

C) =

D) does not exist.

Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

14) Let A be an n × n matrix such that A2 = 0. Then what can we say about the matrix I + A?
A) I + A is invertible, and (I + A)-1 = I + A.
B) I + A is invertible, and (I + A) -1 = I.
C) I + A is invertible, and (I + A) -1 = I – A.
D) I + A is not invertible.
Answer: C
Diff: 3 Type: MC Var: 1
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Find the inverse of a square matrix or show that it is not invertible

Use the inverse to solve a system of equations

Solve the system by using the inverse of the coefficient matrix.
1) 5×1 + 3×2 = 3
2×1 + 5×2 = 24
A) (-3, 6)
B) No solution
C) (6, -3)
D) (-3, -6)
Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Use the inverse to solve a system of equations

2) 6×1 + 4×2 = 4
3×1 = -6
A) (4, -2)
B) (-2, 4)
C) No solution
D) (-2, -4)
Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Use the inverse to solve a system of equations

3) 7×1 – 2×2 = 2
28×1 – 8×2 = 3
A) No solution
B)
C) (4, 4)
D) (2, 3)
Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Use the inverse to solve a system of equations

4) 2×1 + 6×2 = 2
2×1 – x2 = -5
A) (1, -2)
B) (-1, 2)
C) (-2, 1)
D) (2, -1)
Answer: C
Diff: 2 Type: BI Var: 3
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Use the inverse to solve a system of equations
5) 2×1 – 6×2 = -6
3×1 + 2×2 = 13
A) (3, 2)
B) (-2, -3)
C) (2, 3)
D) (-3, -2)
Answer: A
Diff: 2 Type: BI Var: 3
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Use the inverse to solve a system of equations

6) 10×1 – 4×2 = -6
6×1 – x2 = 2
A) (-1, -4)
B) (1, 4)
C) (4, 1)
D) (-4, -1)
Answer: B
Diff: 2 Type: BI Var: 2
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Use the inverse to solve a system of equations

7) 2×1 – 4×2 = -2
3×1 + 4×2 = -23
A) (2, 5)
B) (5, 2)
C) (-5, -2)
D) (-2, 5)
Answer: C
Diff: 2 Type: BI Var: 3
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Use the inverse to solve a system of equations

8) -5×1 + 3 = 8
2×1 – 4 = -20
A) (2, 6)
B) (6, 2)
C) (-6, -2)
D) (-2, -6)
Answer: A
Diff: 2 Type: BI Var: 3
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Use the inverse to solve a system of equations
9) x1 – x2 + 3×3 = -8
2×1 + x3 = 0
x1 + 5×2 + x3 = 40
A) (0, 8, 0)
B) (-8, 0, 0)
C) (8, 8, 0)
D) (0, -8, -8)
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

Determine if a linear mapping is invertible

1) Determine which of the following linear mappings is invertible.
Let fA be a linear mapping whose standard matrix is
A = .

Let T be a linear mapping defined by
T(x1, x2, x3) = (-2×2 – 2×3, -2×1 + 8×2 + 4×3, -x1 – 2×3, 4×2 + 4×3).
A) None
B) Both
C) fA only
D) T only
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (3.5) Inverse Matrices and Inverse Mappings
Skill: Applied
Objective: Determine if a linear mapping is invertible

3.6 Elementary Matrices

Write the elementary matrix for a given row operation

Write the elementary matrix for a given row operation.
1) Write a 4 × 4 elementary matrix that corresponds to swapping the third and fourth rows.
A)

Answer: D
Diff: 1 Type: BI Var: 1
Topic: (3.6) Elementary Matrices
Skill: Applied
Objective: Write the elementary matrix for a given row operation

2) Write a 4 × 4 elementary matrix that corresponds to multiplying the second row by .
A)

Answer: B
Diff: 1 Type: BI Var: 1
Topic: (3.6) Elementary Matrices
Skill: Applied
Objective: Write the elementary matrix for a given row operation

Determine if a matrix is elementary, and if so determine the corresponding row operation

Determine if a matrix is elementary, and if so, determine the corresponding row operation.
1) A =
A) A is the elementary matrix obtained from I4 by mutiplying third row by – .
B) A is not an elementary matrix.
C) A is the elementary matrix obtained from I4 by adding – times the fourth row to the third.
D) A is the elementary matrix obtained from I4 by adding times the fourth row to the third.
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (3.6) Elementary Matrices
Skill: Recall
Objective: Determine if a matrix is elementary, and if so determine the corresponding row operation

Write a matrix and its inverse as a product of elementary matrices

1) Write the matrix A = as a product of elementary matrices.

A) A =

B) A =

C) A =

D) A =

Answer: D
Diff: 3 Type: BI Var: 1
Topic: (3.6) Elementary Matrices
Skill: Applied
Objective: Write a matrix and its inverse as a product of elementary matrices

3.7 LU-Decomposition

Find a LU-Decomposition for a matrix

Find a LU-Decomposition for a matrix.
1) A =

A) A =

B) A =

C) A =

D) A =

Answer: B
Diff: 3 Type: BI Var: 48
Topic: (3.7) LU-Decomposition
Skill: Applied
Objective: Find a LU-Decomposition for a matrix

2) A =

A) A =

B) A =

C) A =

D) A =

Answer: A
Diff: 3 Type: BI Var: 14
Topic: (3.7) LU-Decomposition
Skill: Applied
Objective: Find a LU-Decomposition for a matrix

Solve a system with the LU-Decomposition

Solve the equation Ax = b by using the LU factorization given for A.
1) A = , b =

A =

A) x =

B) x =

C) x =

D) x =

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (3.7) LU-Decomposition
Skill: Applied
Objective: Solve a system with the LU-Decomposition

2) A = , b =

A =

A) x =

B) x =

C) x =

D) x =

Answer: C
Diff: 3 Type: BI Var: 6
Topic: (3.7) LU-Decomposition
Skill: Applied
Objective: Solve a system with the LU-Decomposition