Question 1

Find a nonsingular matrix  such that  is diagonal where  .

Accepted Answer

A)    
B)    
C)    
D)    
E)

Question 2

Use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation  . Identify the resulting rotated conic and give its equation in the new coordinate system.

Accepted Answer

A)  Ellipse;  
B)  Hyperbola;  
C)  Ellipse;  
D)  Hyperbola;  
E)  Ellipse;

Question 3

Find the eigenvalues of the symmetric matrix  . For each eigenvalue, find the dimension of the corresponding eigenspace.

Accepted Answer

A)    
B)    
C)    
D)   
E)

Question 4

Find an orthogonal matrix P such that  diagonalizes  .

Accepted Answer

A)    
B)    
C)    
D)    
E)

Question 5

Compute   given  and  .

Accepted Answer

A)    
B)    
C)    
D)    
E)

Question 6

Find the matrix of the quadratic form associated with the quadratic equation  .

Accepted Answer

A)    
B)    
C)    
D)    
E)

Question 7

Compute  given  and  .

Accepted Answer

A)    
B)    
C)   
D)    
E)

Question 8

The matrix  is diagonalizable.

Accepted Answer

A) True 
 B)False

Question 9

Find the eigenvalues of the symmetric matrix  . For each eigenvalue, find the dimension of the corresponding eigenspace.

Accepted Answer

A)    
B)    
C)   
D)    
E)

Question 10

Find the eigenvalues and corresponding eigenvectors for the matrix  .

Accepted Answer

A)    
B)    
C)    
D)    
E)

Question 11

Which of the following is an eigenvector for the matrix

Accepted Answer

A)    
B)    
C)   
D)    
E)

Question 12

Solve the system of first-order linear differential equations given below.

Accepted Answer

A)   
B)    
C)   
D)    
E)

Exam 7: Eigenvalues Eigenvectors

Find a nonsingular matrix such that is diagonal where .

Use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation . Identify the resulting rotated conic and give its equation in the new coordinate system.

Find the eigenvalues of the symmetric matrix . For each eigenvalue, find the dimension of the corresponding eigenspace.

Find an orthogonal matrix P such that diagonalizes .

Compute given and .

Find the matrix of the quadratic form associated with the quadratic equation .

Compute given and .

The matrix is diagonalizable.

Find the eigenvalues of the symmetric matrix . For each eigenvalue, find the dimension of the corresponding eigenspace.

Find the eigenvalues and corresponding eigenvectors for the matrix .

Which of the following is an eigenvector for the matrix

Solve the system of first-order linear differential equations given below.