Chapter 6 Eigenvectors and Diagonalization- test bank-Introduction to Linear Algebra for Scientists & Engineers

Download file with the answers

Chapter 6 Eigenvectors and Diagonalization- test bank-Introduction to Linear Algebra for Scientists & Engineers
1 file(s) 1.24 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 6 Eigenvectors and Diagonalization

6.1 Eigenvalues and Eigenvectors

Determine if a vector is an eigenvector of a matrix and the corresponding eigenvalue

For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue.
1) A = , λ = 5

Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Recall
Objective: Determine if a vector is an eigenvalue of a matrix and the corresponding eigenvalue

2) A = , λ = 4

Answer: B
Diff: 1 Type: BI Var: 4
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Determine if a vector is an eigenvalue of a matrix and the corresponding eigenvalue

For the given matrix A, find a basis for the corresponding eigenspace for the given eigenvalue.

3) A = , λ = -3

Answer: C
Diff: 2 Type: BI Var: 36
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Determine if a vector is an eigenvector of a matrix and the corresponding eigenvalue

4) A = , λ = -4

Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Determine if a vector is an eigenvector of a matrix and the corresponding eigenvalue
Determine if the vector is an eigenvector of a matrix. If it is, determine the corresponding eigenvalue.

5) A = and =
A) is not an eigen vector.
B) The eigen value is 1.
C) The eigen value is 0.
D) The eigen value is 2.
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Determine if a vector is an eigenvalue of a matrix and the corresponding eigenvalue

6) A = and =
A) is not an eigen vector.
B) The eigen value is 3.
C) The eigen value is 0.
D) The eigen value is 2.
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Determine if a vector is an eigenvalue of a matrix and the corresponding eigenvalue

7) A = and =
A) is not an eigen vector.
B) The eigen value is 3.
C) The eigen value is 0.
D) The eigen value is 2.
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Determine if a vector is an eigenvalue of a matrix and the corresponding eigenvalue

8) A = and =
A) is not an eigen vector.
B) The eigen value is 3.
C) The eigen value is 0.
D) The eigen value is 2.
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Determine if a vector is an eigenvalue of a matrix and the corresponding eigenvalue
9) A = and =
A) is not an eigen vector.
B) The eigen value is 1.
C) The eigen value is 0.
D) The eigen value is 2.
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Determine if a vector is an eigenvalue of a matrix and the corresponding eigenvalue

10) A = and =
A) is not an eigen vector.
B) The eigen value is 3.
C) The eigen value is 1.
D) The eigen value is 2.
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Determine if a vector is an eigenvalue of a matrix and the corresponding eigenvalue

11) A = and =
A) is not an eigen vector.
B) The eigen value is 3.
C) The eigen value is 1.
D) The eigen value is 2.
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Determine if a vector is an eigenvalue of a matrix and the corresponding eigenvalue

Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue

Find the eigenvalues of the given matrix.
1)
A) 1, 2
B) 1
C) 1, -2
D) -2
Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue
2)
A) 3, -5
B) 3
C) -3, 5
D) 5
Answer: C
Diff: 2 Type: BI Var: 50+
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue

Find the characteristic equation of the given matrix.

3) A =
A) (2 – λ)(4 – λ)(8 – λ) = 0
B) (2 – λ)2 (4 – λ)(8 – λ) = 0
C) (2 – λ)(5 – λ)(3 – λ)(1 – λ) = 0
D) (2 – λ)(7 – λ)(8 – λ)(1 – λ) = 0
Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue

4) A =
A) (6 – λ)(5 – λ)(-1 – λ)(9 – λ) = 0
B) (1 – λ)(-7 – λ)(4 – λ)(9 – λ) = 0
C) (9 – λ)(-1 – λ)(5 – λ)(6 – λ) = 0
D) (1 – λ)(-5 – λ)(-7 – λ)(6 – λ) = 0
Answer: D
Diff: 2 Type: BI Var: 50+
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue
The characteristic polynomial of a 5 × 5 matrix is given below. Find the eigenvalues and their multiplicities.

5) λ5 + 17λ4 + 72λ3
A) 0 (multiplicity 2), -9 (multiplicity 1), -8 (multiplicity 1)
B) 0 (multiplicity 3), 8 (multiplicity 1), 9 (multiplicity 1)
C) 0 (multiplicity 3), -9 (multiplicity 1), -8 (multiplicity 1)
D) 0 (multiplicity 1), 8 (multiplicity 1), 9 (multiplicity 1)
Answer: C
Diff: 2 Type: BI Var: 17
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue

6) λ5 – 13λ4 – 9λ3 + 405λ2
A) 0 (multiplicity 2), 9 (multiplicity 2), -5 (multiplicity 1)
B) 0 (multiplicity 2), -9 (multiplicity 2), 5 (multiplicity 1)
C) 0 (multiplicity 1), 9 (multiplicity 3), -5 (multiplicity 1)
D) 0 (multiplicity 2), -9 (multiplicity 2), -5 (multiplicity 1)
Answer: A
Diff: 2 Type: BI Var: 18
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue

Given eigenvalue λ of a matrix A, determine the geometric and algebraic multiplicity of the eigenvalue.

7) A = , λ = 0
A) geometric multiplicity 1 and algebraic multiplicity 2
B) geometric multiplicity 2 and algebraic multiplicity 1
C) geometric multiplicity 2 and algebraic multiplicity 2
D) geometric multiplicity 1 and algebraic multiplicity 1
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue
8) A = , λ = 2
A) geometric multiplicity 1 and algebraic multiplicity 2
B) geometric multiplicity 2 and algebraic multiplicity 1
C) geometric multiplicity 2 and algebraic multiplicity 2
D) geometric multiplicity 1 and algebraic multiplicity 1
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue

9) A = , λ = 5
A) geometric multiplicity 1 and algebraic multiplicity 2
B) geometric multiplicity 2 and algebraic multiplicity 1
C) geometric multiplicity 2 and algebraic multiplicity 2
D) geometric multiplicity 1 and algebraic multiplicity 1
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue

10) A = , λ = 1
A) geometric multiplicity 1 and algebraic multiplicity 2
B) geometric multiplicity 2 and algebraic multiplicity 1
C) geometric multiplicity 2 and algebraic multiplicity 2
D) geometric multiplicity 1 and algebraic multiplicity 1
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue

11) A = where k ≠ 0, λ = 1
A) geometric multiplicity 1 and algebraic multiplicity 2
B) geometric multiplicity 2 and algebraic multiplicity 1
C) geometric multiplicity 2 and algebraic multiplicity 2
D) geometric multiplicity 1 and algebraic multiplicity 1
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue
12) A = , λ = 1
A) geometric multiplicity 1 and algebraic multiplicity 2
B) geometric multiplicity 2 and algebraic multiplicity 1
C) geometric multiplicity 2 and algebraic multiplicity 2
D) geometric multiplicity 1 and algebraic multiplicity 1
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Find the eigenvalues and determine the geometric and algebraic multiplicity of each eigenvalue

Demonstrate understanding of the theory of eigenvalues and eigenvectors

1) Let A be an n × n matrix such that A2 = A. Then the only possible eigen values of A are ________.
A) 1 and -1
B) 0 and -1
C) 0 and 1
D) -1
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Demonstrate understanding of the theory of eigenvalues and eigenvectors

2) Suppose that A is 3 × 3 matrix such that det(A) = 0, det(A – 2I) = 0 and det(A + I) = 0. Then what is the dimension of the solution space of A = ?
A) 0
B) 3
C) 2
D) 1
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Demonstrate understanding of the theory of eigenvalues and eigenvectors

3) Suppose that A is 3 × 3 matrix such that det(A) = 0, det(A – 2I) = 0 and det(A + I) = 0. Then what is the dimension of the null space of A-3I?
A) 0
B) 3
C) 2
D) 1
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Demonstrate understanding of the theory of eigenvalues and eigenvectors
4) Suppose that A is 3 × 3 matrix such that det(A) = 0, det(A – 2I) = 0 and det(A + I) = 0. Then what is the rank of A?
A) 0
B) 3
C) 2
D) 1
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Demonstrate understanding of the theory of eigenvalues and eigenvectors

5) Let A= . Then the eigen values of A are ________.
A) a and b
B) a and c
C) b and c
D) a, b, and c
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Demonstrate understanding of the theory of eigenvalues and eigenvectors

6) Which of the following statements are true for an n × n matrix A?
I) If A has no real eigen values, then A3 has no real eigen values.
II) If the eigen values of A are non zero, then A is invertible.
A) Neither I nor II
B) I only
C) II only
D) Both I and II
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (6.1) Eigenvalues and Eigenvectors
Skill: Applied
Objective: Demonstrate understanding of the theory of eigenvalues and eigenvectors
6.2 Diagonalization

Determine if a matrix P diagonalizes a given matrix A

1) Determine in which case whether P diagonalizes A.
I) A = , P =

II) A = , P =
A) Neither I nor II
B) Both I and II
C) I only
D) II only
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Demonstrate understanding of the theory of eigenvalues and eigenvectors

2) Determine in which case whether P diagonalizes A.
I) A = , P =

II) A = , P =
A) Neither I nor II
B) Both I and II
C) I only
D) II only
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Demonstrate understanding of the theory of eigenvalues and eigenvectors
3) Determine in which case whether P diagonalizes A.
I) A = , P =

II) A = , P =
A) Neither I nor II
B) Both I and II
C) I only
D) II only
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Demonstrate understanding of the theory of eigenvalues and eigenvectors

4) Determine in which case whether P diagonalizes A.
I) A = , P =

II) A = , P =
A) Neither I nor II
B) Both I and II
C) I only
D) II only
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Demonstrate understanding of the theory of eigenvalues and eigenvectors
Determine if a matrix is diagonalizable and if so diagonalize it.

Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that

1) A =

A) P = , D =

B) P = , D =

C) P = , D =

D) P = , D =

Answer: A
Diff: 3 Type: BI Var: 4
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it

2) A =

A) P = , D =
B) P = , D =
C) P = , D =
D) Not diagonizable
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it
3) A =

A) P = , D =

B) Not diagonalizable

C) P = , D =

D) P = , D =

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it

4) A =

A) P = , D =

B) P = , D =

C) Not diagonalizable

D) P = , D =

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it
5) A =

A) P = , D =

B) P = , D =

C) P = , D =

D) Not diagonalizable

Answer: D
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it

6) A =

A) Not diagonalizable

B) P = , D =

C) P = , D =

D) P = , D =

Answer: A
Diff: 3 Type: BI Var: 8
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it
7) A =

A) P = , D =

B) P = , D =

C) P = , D =

D) Not diagonalizable
Answer: A
Diff: 3 Type: BI Var: 7
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it

8) A =

A) P = , D =

B) P = , D =

C) P = , D =

D) Not diagonalizable

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it

9) A =

A) P = , D =

B) P = , D =

C) Not diagonalizable over ℛ.

D) P = , D =

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it

10) A =

A) P = , D =

B) P = , D =

C) Not diagonalizable over ℛ.

D) P = , D =

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it

Define T: ℛ2 → ℛ2 by T(x) = Ax, where A is the matrix defined below. Find the requested basis B for and the corresponding B-matrix for T.

11) Find a basis B for ℛ2 and the B-matrix D for T with the property that D is a diagonal matrix.

A =

A) B = , D =

B) B = , D =

C) B = , D =

D) B = , D =

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it

12) Find a basis B for ℛ2 and the B-matrix D for T with the property that D is an upper triangular matrix.

A =

A) B = , D =

B) B = , D =

C) B = , D =

D) B = , D =

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Determine if a matrix is diagonalizable and if so diagonalize it

Demonstrate knowledge of the theory of diagonalization

1) Let A be a diagonalizable matrix whose eigen values satisfy that λ2 = λ + 1. Then A satisfies ________.
A) A2 = A
B) A2 = A+1
C) A2 = A + I
D) A2 = A + I
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Demonstrate knowledge of the theory of diagonalization

2) Let A be a diagonalizable matrix. If 0 and 1 are the only eigenvalues of A, then ________.
A) A = 0
B) A2 = A
C) A = I
D) A2 = A + I
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Demonstrate knowledge of the theory of diagonalization

3) Let A and B be n × n matrices such that P-1A P = B for some invertible matrix P. Then which of the following statements are true?
I) det(A) = det(B)
II) A and B have the same eigenvectors.
A) Neither I nor II
B) Both I and II
C) II only
D) I only
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Demonstrate understanding of the theory of eigenvalues and eigenvectors

6.3 Powers of Matrices and the Markov Process

Determine if a matrix is a Markov matrix and determine the invariant state

Solve the problem.
1) Suppose that demographic studies show that each year about 6% of a city’s population moves to the suburbs (and 94% stays in the city), while 4% of the suburban population moves to the city (and 96% remains in the suburbs). In the year 2000, 60.7% of the population of the region lived in the city and 39.3% lived in the suburbs. What is the distribution of the population in 2002? For simplicity, ignore other influences on the population such as births, deaths, and migration into and out of the city/suburban region.
A) 55.7% in the city and 44.3% in the suburbs
B) 58.6% in the city and 41.4% in the suburbs
C) 57.7% in the city and 42.3% in the suburbs
D) 56.8% in the city and 43.2% in the suburbs
Answer: D
Diff: 3 Type: BI Var: 50+
Topic: (6.3) Powers of Matrices and the Markov Process
Skill: Applied
Objective: Determine if a matrix is a Markov matrix and determine the invariant state

2) Suppose that the weather in a certain city is either sunny, cloudy, or rainy on a given day, and consider the following:
1) If it is sunny today there is a 70% chance it will be sunny tomorrow and a 30% chance that it will be cloudy.
2) If it is cloudy today there is a 40% chance it will be sunny tomorrow, a 40% chance that it will be cloudy, and a 20% chance that it will be rainy.
3) If it is rainy today there is a 40% chance it will be sunny tomorrow, a 30% chance that it will be cloudy, and a 30% chance that it will be rainy.

Suppose the predicted weather for Friday is 55% sunny, 35% cloudy, and 10% rainy. What are the chances that Sunday will be rainy?
A) 9.7%
B) 10%
C) 11%
D) 8.7%
Answer: A
Diff: 3 Type: BI Var: 11
Topic: (6.3) Powers of Matrices and the Markov Process
Skill: Applied
Objective: Determine if a matrix is a Markov matrix and determine the invariant state

Find the fixed-state vector for the Markov matrix P.

3) P =

Answer: B
Diff: 3 Type: BI Var: 50
Topic: (6.3) Powers of Matrices and the Markov Process
Skill: Recall
Objective: Determine if a matrix is a Markov matrix and determine the invariant state

4) P =

Answer: A
Diff: 3 Type: BI Var: 34
Topic: (6.3) Powers of Matrices and the Markov Process
Skill: Applied
Objective: Determine if a matrix is a Markov matrix and determine the invariant state

Solve the problem.

5) Suppose that demographic studies show that each year about 6% of a city’s population moves to the suburbs (and 94% stays in the city), while 2% of the suburban population moves to the city (and 98% remains in the suburbs). In the year 2000, 61.2% of the population of the region lived in the city and 38.8% lived in the suburbs. What percentage of the population of the region would eventually live in the city if the migration probabilities were to remain constant over many years? For simplicity, ignore other influences on the population such as births, deaths, and migration into and out of the region.
A) 50%
B) 75.0%
C) 25%
D) 37.5%
Answer: C
Diff: 3 Type: BI Var: 50+
Topic: (6.3) Powers of Matrices and the Markov Process
Skill: Applied
Objective: Determine if a matrix is a Markov matrix and determine the invariant state

6) Suppose that the weather in a certain city is either sunny, cloudy, or rainy on a given day, and consider the following:
1) If it is sunny today there is a 70% chance it will be sunny tomorrow and a 30% chance that it will be cloudy.
2) If it is cloudy today there is a 40% chance it will be sunny tomorrow, a 40% chance that it will be cloudy, and a 20% chance that it will be rainy.
3) If it is rainy today there is a 40% chance it will be sunny tomorrow, a 30% chance that it will be cloudy, and a 30% chance that it will be rainy.

In the long run, how likely is it that the weather will be rainy on a given day?
A) 9.5%
B) 10.6%
C) 11.3%
D) 12.1%
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (6.3) Powers of Matrices and the Markov Process
Skill: Applied
Objective: Determine if a matrix is a Markov matrix and determine the invariant state

Use diagonalization to calculate powers of a matrix

Find a formula for Ak, given that A = PDP-1, where P and D are given below.

1) A = , P = , D =

Answer: A
Diff: 3 Type: BI Var: 20
Topic: (6.3) Powers of Matrices and the Markov Process
Skill: Applied
Objective: Use diagonalization to calculate powers of a matrix

2) A = , P = , D =

D)
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (6.3) Powers of Matrices and the Markov Process
Skill: Applied
Objective: Use diagonalization to calculate powers of a matrix

Apply the power method to the matrix A below with x0 = . Stop when k = 5, and determine the dominant eigenvalue and corresponding eigenvector.

3) A =

A) 4,

B) -1,

C) -1,

D) 4,

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (6.3) Powers of Matrices and the Markov Process
Skill: Applied
Objective: Use diagonalization to calculate powers of a matrix
4) A =

A) -4,

B) -4,

C) -3,

D) -3,

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (6.2) Diagonalization
Skill: Applied
Objective: Use diagonalization to calculate powers of a matrix

6.4 Diagonalization and Differential Equations

Find the general solution for a system of linear differential equations

Solve the initial value problem.
1) x’ = Ax, x(0) = , where A = .

A) x(t) = e-4t

B) x(t) = e4t

C) x(t) = e-4t

D) x(t) = e4t

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (6.4) Diagonalization and Differential Equations
Skill: Applied
Objective: Find the general solution for a system of linear differential equations
Find the general solution of the system of linear differential equations.

2) x’ = Ax, where A = .

A) x(t) = ae5t + be4t

B) x(t) = ae-5t + be4t

C) x(t) = ae-5t + be-4t

D) x(t) = ae-5t + be4t

Answer: D
Diff: 3 Type: BI Var: 50+
Topic: (6.4) Diagonalization and Differential Equations
Skill: Applied
Objective: Find the general solution for a system of linear differential equations