Chapter 8 Symmetric Matrices and Quadratic Forms – test bank-Introduction to Linear Algebra for Scientists & Engineers

---

 

Download  file with the answers

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 8 Symmetric Matrices and Quadratic Forms

8.1 Diagonalization of Symmetric Matrices

Determine if a matrix is symmetric

1) Determine which of the following matrices are symmetric.
A = , B =
A) None of them
B) A and B
C) B only
D) A only
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Recall
Objective: Determine if a matrix is symmetric

2) Determine which of the following matrices are symmetric.
A = , B =
A) None of them
B) A and B
C) B only
D) A only
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Recall
Objective: Determine if a matrix is symmetric

3) Which of the following statements are true for an n×n matrix A?
I) ATA is symmetric.
II) If A is symmetric, then A is diagonalizable.
A) None
B) Both
C) I only
D) II only
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Determine if a matrix is symmetric

4) Which of the following statements are true for an n × n matrix A?
I) AT +A is symmetric.
II) If A is symmetric, then all the eigen values of A are real.
A) None
B) Both
C) I only
D) II only
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Determine if a matrix is symmetric

Orthogonally diagonalize a symmetric matrix

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.
1)
A) P = , D =

B) P = , D =

C) P = , D =

D) P = , D =

Answer: D
Diff: 3 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Orthogonally diagonalize a symmetric matrix

2)
A) P = , D =

B) P = , D =

C) P = , D =

D) P = , D =

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Orthogonally diagonalize a symmetric matrix

3)
A) P = , D =

B) P = , D =

C) P = , D =

D) P = , D =

Answer: A
Diff: 3 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Orthogonally diagonalize a symmetric matrix

4)
A) P = , D =

B) P = , D =

C) P = , D =

D) P = , D =

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Orthogonally diagonalize a symmetric matrix

5)
A) P = , D =

B) P = , D =

C) P = , D =

D) P = , D =

Answer: D
Diff: 3 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Orthogonally diagonalize a symmetric matrix

6)
A) P = , D =

B) P = , D =

C) P = , D =

D) P = , D =

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Orthogonally diagonalize a symmetric matrix

7)
A) P = , D =

B) P = , D =

C) P = , D =

D) P = , D =

Answer: D
Diff: 3 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Orthogonally diagonalize a symmetric matrix

8)
A) P = , D =

B) P = , D =

C) P = , D =

D) P = , D =

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Orthogonally diagonalize a symmetric matrix

9)
A) P = , D =

B) P = , D =

C) P = , D =

D) P = , D =

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Orthogonally diagonalize a symmetric matrix

10)
A) P = , D =

B) P = , D =

C) P = , D =

D) P = , D =

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (8.1) Diagonalization of Symmetric Matrices
Skill: Applied
Objective: Orthogonally diagonalize a symmetric matrix

8.2 Quadratic Forms

Change between quadratic forms and their associated symmeteric matrix

Compute the quadratic form A for the given matrix A and vector .
1) A = , =
A) -6 – 3
B) -3 – 6
C) -3×1 – 6×2
D) -6×1 – 12×2
Answer: B
Diff: 1 Type: BI Var: 50+
Topic: (8.2) Quadratic Forms
Skill: Recall
Objective: Change between quadratic forms and their associated symmetric matrix
2) A = , =
A) 5 + 3
B) 3 + 3
C) 3×1 + 3×2
D) 3×12 + 5x1x2 + 2×22
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Recall
Objective: Change between quadratic forms and their associated symmetric matrix

3) A = , =
A) 9 + – 2 + 6x1x2 + 10x2x3
B) 9 + – 2
C) 9 – + 2 – 6x1x2 – 10x2x3
D) 9 + – 2 + 3×1 + 8×2 + 5×3
Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (8.2) Quadratic Forms
Skill: Recall
Objective: Change between quadratic forms and their associated symmetric matrix
4) A = , =
A) 3 + – 2 + 6x1x2 + 10x2x3
B) 3 + – 2 + x1x2 + 4x2x3 + 2x1x3
C) 3 + x1x2 + 4x2x3 + 2x1x3
D) 3 + x1x2 + 4x2x3 – 2x1x3
Answer: C
Diff: 1 Type: BI Var: 50+
Topic: (8.2) Quadratic Forms
Skill: Recall
Objective: Change between quadratic forms and their associated symmetric matrix

Find the matrix of the quadratic form.
5) 3 + 5 + 2 – 12x1x2 + x2x3
A)

B)

C)

D)

Answer: B
Diff: 1 Type: BI Var: 50+
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Change between quadratic forms and their associated symmetric matrix
6) 8x1x2 – 16x1x3 + 6x2x3
A)

B)

C)

D)

Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Change between quadratic forms and their associated symmetric matrix

7) x12 + 4x1x2 + 2 x22
A)

B)

C)

D)

Answer: D
Diff: 1 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Recall
Objective: Change between quadratic forms and their associated symmetric matrix
8) -2x1x2
A)

B)

C)

D)

Answer: C
Diff: 1 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Recall
Objective: Change between quadratic forms and their associated symmetric matrix

9) 3 + 5 + 2
A)

B)

C)

D)

Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Change between quadratic forms and their associated symmetric matrix
10) + + + 2x1x2 + 2x2x3
A)

B)

C)

D)

Answer: C
Diff: 1 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Change between quadratic forms and their associated symmetric matrix

Bring a quadratic form into diagonal form and give the change of variables

Make a change of variable, = P , that transforms the given quadratic form into a quadratic form with no cross-product term. Give P and the new quadratic form.
1) Q( ) = 4 + 7 + 4x1x2
A) P = ; -8 – 3
B) P = ; 8 + 3
C) P = ; 8y1 + 3y2
D) P = ; 8 + 3
Answer: D
Diff: 3 Type: BI Var: 3
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Bring a quadratic form into diagonal form and give the change of variables
2) Q( ) = + + 4x1x2
A) P = ; 3 –
B) P = ; 3 +
C) P = ; 3y1 – y2
D) P = ; 3y1 + y2
Answer: A
Diff: 3 Type: BI Var: 3
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Bring a quadratic form into diagonal form and give the change of variables

3) Q( ) = 5 + 2 + 4x1x2
A) P = ; 6 +
B) P = ; 6 +
C) P = ; 6y1 + y2
D) P = ; 6 –
Answer: B
Diff: 3 Type: BI Var: 3
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Bring a quadratic form into diagonal form and give the change of variables
4) Q( ) = 13 + 12 + 5 + 20x1x2 + 8x1x3 + 12x2x3
A) P = ; 25 + 4 +
B) P = ; 25 + 4 +
C) P = ; + 4 + 25
D) P = ; 5 + 2 +
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Bring a quadratic form into diagonal form and give the change of variables

Classify a quadratic form and/or symmetric matrix

Classify the given quadratic form as semidefinite, indefinite, negative definite, or positive definite.
1) Q( ) = 4 + 7 + 4x1x2
A) semidefinite
B) indefinite
C) negative definite
D) positive definite
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

2) Q( ) = + + 4x1x2
A) semidefinite
B) indefinite
C) negative definite
D) positive definite
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix
3) Q( ) = 13 + 12 + 5 + 20x1x2 + 8x1x3 + 12x2x3
A) semidefinite
B) indefinite
C) negative definite
D) positive definite
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

4) Q( ) = -4 – 5 -4 + 2x1x2 – 2x2x3
A) semidefinite
B) indefinite
C) negative definite
D) positive definite
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

5) Q( ) = 4 + +
A) semidefinite
B) indefinite
C) negative definite
D) positive definite
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

6) Q( ) = -4 – –
A) semidefinite
B) indefinite
C) negative definite
D) positive definite
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix
7) Q( ) = – –
A) semidefinite
B) indefinite
C) negative definite
D) positive definite
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

8) Q( ) = 4 + 3 + 8x1x2
A) semidefinite
B) indefinite
C) negative definite
D) positive definite
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

Classify the given symmetric matrix as indefinite, semidefinite, positive definite, or negative definite.

9)
A) indefinite
B) semidefinite
C) positive definite
D) negative definite
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

10)
A) indefinite
B) semidefinite
C) positive definite
D) negative definite
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix
11)
A) indefinite
B) semidefinite
C) positive definite
D) negative definite
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

12)
A) indefinite
B) semidefinite
C) positive definite
D) negative definite
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

13)
A) indefinite
B) semidefinite
C) positive definite
D) negative definite
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

14)
A) indefinite
B) semidefinite
C) positive definite
D) negative definite
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix
15)
A) indefinite
B) semidefinite
C) positive definite
D) negative definite
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

16)
A) indefinite
B) positive semidefinite
C) positive definite
D) negative definite
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

17)
A) indefinite
B) negative semidefinite
C) positive definite
D) negative definite
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

18)
A) indefinite
B) positive semidefinite
C) positive definite
D) negative definite
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (8.2) Quadratic Forms
Skill: Applied
Objective: Classify a quadratic form and/or symmetric matrix

8.3 Graphs of Quadratic Forms

Sketch the graph of a quadratic form on R2

Sketch the graph of the equation on R2. Show both the original axes and the new axes.
1) 5×2 + 2y2 + xy = 6
A)

B)

C)

D)

Answer: C
Diff: 2 Type: BI Var: 1
Topic: (8.3) Graphs of Quadratic Forms
Skill: Applied
Objective: Sketch the graph of a quadratic form on R2

2) 5 – 3 + 6x1x2 = 15
A)

B)

C)

D)

Answer: C
Diff: 2 Type: BI Var: 1
Topic: (8.3) Graphs of Quadratic Forms
Skill: Applied
Objective: Sketch the graph of a quadratic form on R2

Identify the shape of the graph of a quadratic form on R2 or R3

Identify the shape of the graph of the quadratic form on R2.
1) 2 + 2 – 3x1x2 = 6
A) Set of two parallel lines
B) Circle
C) Hyperbola
D) Ellipse
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (8.3) Graphs of Quadratic Forms
Skill: Applied
Objective: Identify the shape of the graph of a quadratic form on R2 or R3
2) 4 + 9 = 36
A) Set of two parallel lines
B) Circle
C) Hyperbola
D) Ellipse
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (8.3) Graphs of Quadratic Forms
Skill: Applied
Objective: Identify the shape of the graph of a quadratic form on R2 or R3

3) 4 + + 4x1x2 = 1
A) Set of two parallel lines
B) Circle
C) Hyperbola
D) Ellipse
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (8.3) Graphs of Quadratic Forms
Skill: Applied
Objective: Identify the shape of the graph of a quadratic form on R2 or R3

4) 5 – 3 + 9x1x2 = -1
A) Set of two parallel lines
B) Circle
C) Hyperbola
D) Ellipse
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (8.3) Graphs of Quadratic Forms
Skill: Applied
Objective: Identify the shape of the graph of a quadratic form on R2 or R3

Identify the shape of the graph of the quadratic form on ℛ3.
5) – – 4x2x3 – 4x1x3 = -1
A) Ellipsoid
B) Elliptic cylinder
C) Hyperbolic cylinder
D) Hyperboloid of two sheet
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (8.3) Graphs of Quadratic Forms
Skill: Applied
Objective: Identify the shape of the graph of a quadratic form on R2 or R3
6) + 2 = 1
A) Ellipsoid
B) Elliptic cylinder
C) Hyperbolic cylinder
D) Hyperboloid of two sheet
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (8.3) Graphs of Quadratic Forms
Skill: Applied
Objective: Identify the shape of the graph of a quadratic form on R2 or R3

7) 2x1x2 + 2x2x3 + 2 x1x3 = 1
A) Ellipsoid
B) Elliptic cylinder
C) Hyperbolic cylinder
D) Hyperboloid of two sheet
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (8.3) Graphs of Quadratic Forms
Skill: Applied
Objective: Identify the shape of the graph of a quadratic form on R2 or R3

8.4 Applications of Quadratic Forms

Applications of Quadratic Forms

Find the maximum value of Q(x) subject to the constraint x = 1.
1) Q(x) = 2 + 7 + 5
A) 2
B) 7
C) 5
D) 14
Answer: B
Diff: 3 Type: BI Var: 16
Topic: (8.4) Applications of Quadratic Forms
Skill: Applied
Objective: Applications of quadratic forms

2) Q(x) = 14 + 14 + 18 + 26x1x2 + 18x1x3 + 18x2x3
A) 9
B) 1
C) 36
D) 25
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (8.4) Applications of Quadratic Forms
Skill: Applied
Objective: Applications of quadratic forms

Find a unit vector at which the quadratic form Ax is maximized, subject to the constraint x = 1.

3) A =
A)
B)
C)
D)
Answer: A
Diff: 3 Type: BI Var: 3
Topic: (8.3) Graphs of Quadratic Forms
Skill: Applied
Objective: Applications of quadratic forms

4) A =
A)
B)
C)
D)
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (8.4) Applications of Quadratic Forms
Skill: Applied
Objective: Applications of quadratic forms

Find the maximum value of Q(x) subject to the constraints x = 1 and u = 0, where u is a unit eigenvector corresponding to the greatest eigenvalue of the matrix of the quadratic form.

5) Q(x) = 2 + 7 + 4
A) 7
B) 2
C) 4
D) 0
Answer: C
Diff: 3 Type: BI Var: 16
Topic: (8.4) Applications of Quadratic Forms
Skill: Applied
Objective: Sketch the graph of a quadratic form on R2

6) Q(x) = 2 + 7 + 4
A) 4
B) 2
C) 7
D) 0
Answer: A
Diff: 3 Type: BI Var: 16
Topic: (8.4) Applications of Quadratic Forms
Skill: Applied
Objective: Applications of quadratic forms