Question 1

Diagonalize the matrix A, if possible.

Accepted Answer

where 
  
 and

Question 2

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

Accepted Answer

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

Question 3

Diagonalize the given matrix A, and use the diagonalization to compute  
.

Accepted Answer

The answer of Diagonalize the given matrix A, and use...

Question 4

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 5

Compute the first two iterations of the Power Method with scaling, starting with the given  , rounding any numerical values to two decimal places.

Accepted Answer

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

Question 6

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  
. If A is invertible, then  is a solution to the system  
.

Accepted Answer

A) True 
 B)False

Question 7

Suppose  is a  matrix having eigenvalues  
. Then  has dominant eigenvalue  
.

Accepted Answer

A) True 
 B)False

Question 8

If  , and  are solutions to the initial-value problem  , with  , then  for all t.

Accepted Answer

A) True 
 B)False

Question 9

If the  invertible matrix A has hidden rotation-dilation matrix  , where  then  has hidden rotation-dilation matrix  
.

Accepted Answer

A) True 
 B)False

Question 10

The Shifted Inverse Power Method for an invertible n×nmatrix Ais implemented by applying the Power Method to  for some scalar c.

Accepted Answer

A) True 
 B)False

Question 11

Find the rotation-dilation matrix B within the given matrix A.

Accepted Answer

The answer of Find the rotation-dilation matrix B within the...

Question 12

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 13

Compute the first two iterations of the Power Method with scaling, starting with the given  , rounding any numerical values to two decimal places.

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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 16

Diagonalize the given matrix A, and use the diagonalization to compute  
.

Accepted Answer

The answer of Diagonalize the given matrix A, and use...

Question 17

Compute  if  
.

Accepted Answer

The answer of Compute  if  
.
 ...

Question 18

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 19

Determine the rotation and dilation that result from multiplying vectors in  by the given matrix.

Accepted Answer

The answer of Determine the rotation and dilation that result...

Question 20

Determine which of  ,  , and  are eigenvectors of  
,
and determine the associated eigenvalues.

Accepted Answer

The answer of Determine which of  ,  ,...

Exam 6: Eigenvalues and Eigenvectors

Diagonalize the matrix A, if possible.

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

Diagonalize the given matrix A, and use the diagonalization to compute .

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.

Compute the first two iterations of the Power Method with scaling, starting with the given , rounding any numerical values to two decimal places.

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 . If A is invertible, then is a solution to the system .

Suppose is a matrix having eigenvalues . Then has dominant eigenvalue .

If , and are solutions to the initial-value problem , with , then for all t.

If the invertible matrix A has hidden rotation-dilation matrix , where then has hidden rotation-dilation matrix .

The Shifted Inverse Power Method for an invertible n×nmatrix Ais implemented by applying the Power Method to for some scalar c.

Find the rotation-dilation matrix B within the given matrix A.

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

Compute the first two iterations of the Power Method with scaling, starting with the given , rounding any numerical values to two decimal places.

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

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

Diagonalize the given matrix A, and use the diagonalization to compute .

Compute if .

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

Determine the rotation and dilation that result from multiplying vectors in by the given matrix.

Determine which of , , and are eigenvectors of , and determine the associated eigenvalues.