Question 1

The matrix  is diagonalizable.

Accepted Answer

A) True 
 B)False

Question 2

Suppose that A is an  matrix and  is a solution to the system of linear differential equations  where  is an eigenvector of A with associated eigenvalue  
. Let k be any scalar. Then  is a solution to the system  
.

Accepted Answer

A) True 
 B)False

Question 3

The Power Method applied to the matrix  and vector  converges.

Accepted Answer

A) True 
 B)False

Question 4

If  is an eigenvalue of an invertible  matrix A, then  is an eigenvalue of the matrix  
.

Accepted Answer

A) True 
 B)False

Question 5

Factor the matrix  from Question 1 in the form  where B is a rotation-dilation matrix.

Accepted Answer

The answer of Factor the matrix  from Question 1...

Question 6

If A is diagonalizable, then  is diagonalizable.

Accepted Answer

A) True 
 B)False

Question 7

Find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A =  
.

Accepted Answer

The answer of Find the characteristic polynomial, the eigenvalues, and...

Question 8

Diagonalize the matrix A, if possible.

Accepted Answer

The answer of Diagonalize the matrix A, if possible.
 ...

Question 9

Compute the first two iterations of the Shifted Inverse Power Method, starting with the given  , to determine the eigenvalue of A closest to  , rounding any numerical values to two decimal places.

Accepted Answer

The answer of Compute the first two iterations of the...

Question 10

If  is the characteristic polynomial of an  matrix A, and  , then A is not invertible.

Accepted Answer

A) True 
 B)False

Question 11

The coefficient matrix for a system of linear differential equations of the form  has the given eigenvalues and eigenspace bases. Find the general solution for the system.

Accepted Answer

The answer of The coefficient matrix for a system of...

Question 12

The Inverse Power Method applied to the matrix  and vector  converges.

Accepted Answer

A) True 
 B)False

Question 13

The coefficient matrix for a system of linear differential equations of the form  has the given eigenvalues and eigenspace bases. Find the general solution for the system.

Accepted Answer

The answer of The coefficient matrix for a system of...

Question 14

Find the matrix A that has the given eigenvalues and corresponding eigenvectors.

Accepted Answer

The answer of Find the matrix A that has the...

Question 15

Suppose that two countries are in an arms race modeled by the system of differential equations where y₁ and y₂ are measured in thousands. Find the solution for the system with initial conditions , and use it to predict the long-term amounts of arms held by each country.

Accepted Answer

The answer of Suppose that two countries are in an...

Question 16

If  is an eigenvalue of the real  matrix A with corresponding eigenvector  , then  is an eigenvalue of  with corresponding eigenvector  
.

Accepted Answer

A) True 
 B)False

Question 17

Find the eigenvalues and a basis for each eigenspace for the given matrix.

Accepted Answer

The answer of Find the eigenvalues and a basis for...

Question 18

Find a basis for the eigenspace associated with eigenvalue  for matrix  
.

Accepted Answer

The answer of Find a basis for the eigenspace associated...

Question 19

Suppose the  matrix A has n distinct eigenvalues. Then the dimension of each eigenspace is 1.

Accepted Answer

A) True 
 B)False

Question 20

The Power Method applied to the matrix  and vector  converges.

Accepted Answer

A) True 
 B)False

Exam 6: Eigenvalues and Eigenvectors

The matrix is diagonalizable.

Suppose that A is an matrix and is a solution to the system of linear differential equations where is an eigenvector of A with associated eigenvalue . Let k be any scalar. Then is a solution to the system .

The Power Method applied to the matrix and vector converges.

If is an eigenvalue of an invertible matrix A, then is an eigenvalue of the matrix .

Factor the matrix from Question 1 in the form where B is a rotation-dilation matrix.

If A is diagonalizable, then is diagonalizable.

Find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A = .

Diagonalize the matrix A, if possible.

Compute the first two iterations of the Shifted Inverse Power Method, starting with the given , to determine the eigenvalue of A closest to , rounding any numerical values to two decimal places.

If is the characteristic polynomial of an matrix A, and , then A is not invertible.

The coefficient matrix for a system of linear differential equations of the form has the given eigenvalues and eigenspace bases. Find the general solution for the system.

The Inverse Power Method applied to the matrix and vector converges.

The coefficient matrix for a system of linear differential equations of the form has the given eigenvalues and eigenspace bases. Find the general solution for the system.

Find the matrix A that has the given eigenvalues and corresponding eigenvectors.

Suppose that two countries are in an arms race modeled by the system of differential equations where y1 and y2 are measured in thousands. Find the solution for the system with initial conditions , and use it to predict the long-term amounts of arms held by each country.

If is an eigenvalue of the real matrix A with corresponding eigenvector , then is an eigenvalue of with corresponding eigenvector .

Find the eigenvalues and a basis for each eigenspace for the given matrix.

Find a basis for the eigenspace associated with eigenvalue for matrix .

Suppose the matrix A has n distinct eigenvalues. Then the dimension of each eigenspace is 1.

The Power Method applied to the matrix and vector converges.

Suppose that two countries are in an arms race modeled by the system of differential equations where y₁ and y₂ are measured in thousands. Find the solution for the system with initial conditions , and use it to predict the long-term amounts of arms held by each country.