Question 1

Which of the following are eigenvalues with corresponding eigenvectors for the matrix

Accepted Answer

A)

B)

C)  
  
D)  
  
E)  
  
A)

B)

C)  
  
D)  
  
E)  
  A

Question 2

Find the eigenvalues and corresponding eigenvectors for the matrix if the characteristic equation of the matrix is   if the characteristic equation of the matrix is   .

Accepted Answer

A)     
B)     
C)    
D)     
E)    
A)     
B)     
C)    
D)     
E)    C

Question 3

Find an orthogonal matrix P such that   diagonalizes   .

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)     E

Question 4

Find a stable age distribution for the age transition matrix   .

Accepted Answer

A)     , where t is any positive number 
B)     , where t is any positive number 
C)     , where t is any positive number 
D)     , where t is any positive number 
E)     , where t is any positive number 
A)     , where t is any positive number 
B)     , where t is any positive number 
C)     , where t is any positive number 
D)     , where t is any positive number 
E)     , where t is any positive number

Question 5

Find a nonsingular matrix   such that   is diagonal where   .

Accepted Answer

A)     
B)     
C)     
D)     
E)    
A)     
B)     
C)     
D)     
E)

Question 6

The matrix   is diagonalizable.

Accepted Answer

A) True 
 B)False

Question 7

Find the characteristic equation of the matrix   .

Accepted Answer

A)     
B)     
C)     
D)     
E)    
A)     
B)     
C)     
D)     
E)

Question 8

The matrix   is orthogonal.

Accepted Answer

A) True 
 B)False

Question 9

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

Accepted Answer

A)    
B)    
C)     
D)     
E)     
A)    
B)    
C)     
D)     
E)

Question 10

Find the characteristic equation of the matrix   .

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 11

Find a basis   for the domain of   such that the matrix of   relative to   is diagonal.

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 12

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

Question 13

The matrix   is symmetric.

Accepted Answer

A) True 
 B)False

Question 14

Use the age transition matrix   and age distribution vector   to find the age distribution vectors   and   .

Accepted Answer

A)     and   
B)     and   
C)     and   
D)     and   
E)     and   
A)     and   
B)     and   
C)     and   
D)     and   
E)     and

Question 15

Find the eigenvalues and corresponding eigenvectors for the matrix   if the characteristic equation of the matrix is   .

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 16

Find the eigenvalues and corresponding eigenvectors for the matrix   .

Accepted Answer

A)     
B)     
C)     
D)     
E)    
A)     
B)     
C)     
D)     
E)

Question 17

Find a stable age distribution for the age transition matrix   .

Accepted Answer

A)     , where t is any positive number 
B)     , where t is any positive number 
C)     , where t is any positive number 
D)     , where t is any positive number 
E)     , where t is any positive number 
A)     , where t is any positive number 
B)     , where t is any positive number 
C)     , where t is any positive number 
D)     , where t is any positive number 
E)     , where t is any positive number

Question 18

Which of the following is an eigenvector for the matrix

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 19

Suppose a population has the characteristics listed below.
(a) A total of 60% of the population survives its first year. Of that 60%, 50% survives the second year. The maximum life span is 3 years.
(b) The average number of offspring for each member of the population is 4 the first year, 5 the second year, and 4 the third year.The population now consists of 150 members in each of the three age class. How many members will be in each age class after one year? After two years? Let   , and   be vectors whose components are the number members in each age class after one year and after two years respectively.

Accepted Answer

A)     and   
B)    and   
C)     and   
D)     and   
E)     and   
A)     and   
B)    and   
C)     and   
D)     and   
E)     and

Question 20

The matrix   is orthogonal.

Accepted Answer

A) True 
 B)False

Which of the following are eigenvalues with corresponding eigenvectors for the matrix

Find the eigenvalues and corresponding eigenvectors for the matrix if the characteristic equation of the matrix is if the characteristic equation of the matrix is .

Find an orthogonal matrix P such that diagonalizes .

Find a stable age distribution for the age transition matrix .

Find a nonsingular matrix such that is diagonal where .

The matrix is diagonalizable.

Find the characteristic equation of the matrix .

The matrix is orthogonal.

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

Find the characteristic equation of the matrix .

Find a basis for the domain of such that the matrix of relative to is diagonal.

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.

The matrix is symmetric.

Use the age transition matrix and age distribution vector to find the age distribution vectors and .

Find the eigenvalues and corresponding eigenvectors for the matrix if the characteristic equation of the matrix is .

Find the eigenvalues and corresponding eigenvectors for the matrix .

Find a stable age distribution for the age transition matrix .

Which of the following is an eigenvector for the matrix

The matrix is orthogonal.

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Exam 7: Eigenvalues Eigenvectors

Which of the following are eigenvalues with corresponding eigenvectors for the matrix

Find the eigenvalues and corresponding eigenvectors for the matrix if the characteristic equation of the matrix is if the characteristic equation of the matrix is .

Find an orthogonal matrix P such that diagonalizes .

Find a stable age distribution for the age transition matrix .

Find a nonsingular matrix such that is diagonal where .

The matrix is diagonalizable.

Find the characteristic equation of the matrix .

The matrix is orthogonal.

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

Find the characteristic equation of the matrix .

Find a basis for the domain of such that the matrix of relative to is diagonal.

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.

The matrix is symmetric.

Use the age transition matrix and age distribution vector to find the age distribution vectors and .

Find the eigenvalues and corresponding eigenvectors for the matrix if the characteristic equation of the matrix is .

Find the eigenvalues and corresponding eigenvectors for the matrix .

Find a stable age distribution for the age transition matrix .

Which of the following is an eigenvector for the matrix

The matrix is orthogonal.

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Matrices

Determinants

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Inner Product Spaces

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