Deck 7: Eigenvalues Eigenvectors

Question 1

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

A)


B)


C)

D)

E)

Answer 1

Question 2

Which of the following is an eigenvector for the matrix

A)
B)
C)
D)
E)

Answer 2

Question 3

Which of the following is an eigenvector for the matrix

A)
B)
C)
D)
E)

Answer 3

Question 4

Find the characteristic equation of the matrix .

A)
B)
C)
D)
E)

Question 5

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

A)
B)
C)
D)
E)

Question 6

Find the eigenvalues and corresponding eigenvectors for the matrix .

A)
B)
C)
D)
E)

Question 7

Find the characteristic equation of the matrix .

A)
B)
C)
D)
E)

Question 8

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 .

A)
B)
C)
D)
E)

Question 9

Find the eigenvalues and corresponding eigenvectors for the matrix .

A)
B)
C)
D)
E)

Question 10

Compute given and .

A)
B)
C)
D)
E)

Question 11

Compute given and .

A)
B)
C)
D)
E)

Question 12

The matrix is diagonalizable.

Question 13

The matrix is diagonalizable.

Question 14

Find a nonsingular matrix such that is diagonal where .

A)
B)
C)
D)
E)

Question 15

Find a nonsingular matrix such that is diagonal where .

A)
B)
C)
D)
E)

Question 16

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

A)
B)
C)
D)
E)

Question 17

The matrix is symmetric.

Question 18

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

A)
B)
C)
D)
E)

Question 19

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

A)
B)
C)
D)
E)

Question 20

The matrix is orthogonal.

Question 21

The matrix is orthogonal.

Question 22

Find an orthogonal matrix P such that diagonalizes .

A)
B)
C)
D)
E)

Question 23

Find an orthogonal matrix P such that diagonalizes .

A)
B)
C)
D)
E)

Question 24

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

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

Question 25

Find a stable age distribution for the age transition matrix .

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 26

Find a stable age distribution for the age transition matrix .

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 27

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.

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

Question 28

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

A)
B)
C)
D)
E)

Question 29

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

A)
B)
C)
D)
E)

Question 30

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

A)
B)
C)
D)
E)

Question 31

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.

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

Question 32

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.

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