Question 1

Let A and B be square matrices of order 3 such that   and   . Find   . Round your answer to two decimal places when appropriate.

Accepted Answer

A)  0.02 
B)  56 
C)  0.88 
D)  15 
E)  1 
A)  0.02 
B)  56 
C)  0.88 
D)  15 
E)  1

Question 2

Which property of determinants is illustrated by the equation

Accepted Answer

A)  If two rows of a matrix are interchanged, the determinant of the matrix is multiplied by -1. 
B)  If two columns of a matrix are interchanged, the determinant of the matrix is multiplied by -1. 
C)  If a row of a matrix is subtracted from another row of the same matrix, the determinant of the matrix is multiplied by -1. 
D)  If two matrices have any rows equal, the determinant of the matrix is multiplied by -1. 
E)  If a column of a matrix is subtracted from another column, the determinant of the matrix is multiplied by -1. 
A)  If two rows of a matrix are interchanged, the determinant of the matrix is multiplied by -1. 
B)  If two columns of a matrix are interchanged, the determinant of the matrix is multiplied by -1. 
C)  If a row of a matrix is subtracted from another row of the same matrix, the determinant of the matrix is multiplied by -1. 
D)  If two matrices have any rows equal, the determinant of the matrix is multiplied by -1. 
E)  If a column of a matrix is subtracted from another column, the determinant of the matrix is multiplied by -1.

Question 3

Evaluate   for the matrix   .

Accepted Answer

A)  40 
B)  -40 
C)  -400 
D)  400 
E)  80 
A)  40 
B)  -40 
C)  -400 
D)  400 
E)  80

Question 4

Find the determinant of   .

Accepted Answer

A)  -14 
B)  30 
C)  22 
D)  -6 
E)  -18 
A)  -14 
B)  30 
C)  22 
D)  -6 
E)  -18

Question 5

The eigenvalues of the matrix   are   and   . Find a non-zero eigenvector associated with each eigenvalue.

Accepted Answer

A)     ,   ;   ,   
B)     ,   ;   ,   
C)     ,   ;   ,   
D)     ,   ;   ,   
E)    ,   ;   ,   
A)     ,   ;   ,   
B)     ,   ;   ,   
C)     ,   ;   ,   
D)     ,   ;   ,   
E)    ,   ;   ,

Question 6

Use the determinant to decide whether the matrix given below is singular or nonsingular.

Accepted Answer

A)  nonsingular 
B)  singular 
A)  nonsingular 
B)  singular

Question 7

Find the characteristic equation of the matrix   .

Accepted Answer

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

Question 8

Use elementary row or column operations to evaluate the determinant.

Accepted Answer

A)  -84 
B)  1,812 
C)  -1,212 
D)  180 
E)  -180 
A)  -84 
B)  1,812 
C)  -1,212 
D)  180 
E)  -180

Question 9

Find the eigenvalues of the matrix   .

Accepted Answer

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

Question 10

Evaluate   for the matrix   .

Accepted Answer

A)  7 
B)  -21 
C)     
D)     
E)  -7 
A)  7 
B)  -21 
C)     
D)     
E)  -7

Question 11

Use the fact that for an   matrix A,   , to evaluate   .

Accepted Answer

A)  -120 
B)  -360 
C)  -1,000 
D)  -40 
E)  -2,000 
A)  -120 
B)  -360 
C)  -1,000 
D)  -40 
E)  -2,000

Question 12

The eigenvalues of the matrix   are   ,   ,   . Find a non-zero eigenvector corresponding to each of the given eigenvalues.

Accepted Answer

A)   ,   ;   ,   ;   ,   
B)     ,   ;   ,   ;   ,   
C)    ,   ;   ,   ;   ,   
D)    ,   ;   ,   ;   ,   
E)    ,   ;   ,   ;   ,   
A)   ,   ;   ,   ;   ,   
B)     ,   ;   ,   ;   ,   
C)    ,   ;   ,   ;   ,   
D)    ,   ;   ,   ;   ,   
E)    ,   ;   ,   ;   ,

Question 13

Find the eigenvalues of the matrix   .

Accepted Answer

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

Question 14

Which property of determinants is illustrated by the equation

Accepted Answer

A)  If a multiple of a row is added to another row in a matrix and then the constant is added to a row, the determinant of the matrix is unchanged. 
B)  If a non-zero constant is added to a row of a matrix, the determinant of the matrix is unchanged. 
C)  If the two columns of the matrix are interchanged, the determinant of the matrix is unchanged. 
D)  If a multiple of a row is added to another row of a matrix, the determinant of the matrix is unchanged. 
E)  If times a row of a matrixis added to another row of a matrix, the determinant of the matrix is multiplied by c.   
A)  If a multiple of a row is added to another row in a matrix and then the constant is added to a row, the determinant of the matrix is unchanged. 
B)  If a non-zero constant is added to a row of a matrix, the determinant of the matrix is unchanged. 
C)  If the two columns of the matrix are interchanged, the determinant of the matrix is unchanged. 
D)  If a multiple of a row is added to another row of a matrix, the determinant of the matrix is unchanged. 
E)  If times a row of a matrixis added to another row of a matrix, the determinant of the matrix is multiplied by c.

Question 15

Find the minor   and its cofactor   of the matrix   .

Accepted Answer

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

Question 16

Find the determinant of the elementary matrix   .

Accepted Answer

A)  -1 
B)  1 
C)  -2 
D)  0 
E)  2 
A)  -1 
B)  1 
C)  -2 
D)  0 
E)  2

Question 17

Find the characteristic equation of the matrix   .

Accepted Answer

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

Question 18

Find the eigenvalues of the matrix   . If an eigenvalue has multiplicity greater than one, list the eigenvalue according to its multiplicity. For example, list an eigenvalue with multiplicity two twice.

Accepted Answer

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

Question 19

Solve for x given the following equation involving a determinant.

Accepted Answer

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

Question 20

Determine whether the points   ,   , and   are collinear.

Accepted Answer

A) True 
 B)False

Let A and B be square matrices of order 3 such that and . Find . Round your answer to two decimal places when appropriate.

Which property of determinants is illustrated by the equation

Evaluate for the matrix .

Find the determinant of .

The eigenvalues of the matrix are and . Find a non-zero eigenvector associated with each eigenvalue.

Use the determinant to decide whether the matrix given below is singular or nonsingular.

Find the characteristic equation of the matrix .

Use elementary row or column operations to evaluate the determinant.

Find the eigenvalues of the matrix .

Evaluate for the matrix .

Use the fact that for an matrix A, , to evaluate .

The eigenvalues of the matrix are , , . Find a non-zero eigenvector corresponding to each of the given eigenvalues.

Find the eigenvalues of the matrix .

Which property of determinants is illustrated by the equation

Find the minor and its cofactor of the matrix .

Find the determinant of the elementary matrix .

Find the characteristic equation of the matrix .

Find the eigenvalues of the matrix . If an eigenvalue has multiplicity greater than one, list the eigenvalue according to its multiplicity. For example, list an eigenvalue with multiplicity two twice.

Solve for x given the following equation involving a determinant.

Determine whether the points , , and are collinear.

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Exam 3: Determinants

Let A and B be square matrices of order 3 such that and . Find . Round your answer to two decimal places when appropriate.

Which property of determinants is illustrated by the equation

Evaluate for the matrix .

Find the determinant of .

The eigenvalues of the matrix are and . Find a non-zero eigenvector associated with each eigenvalue.

Use the determinant to decide whether the matrix given below is singular or nonsingular.

Find the characteristic equation of the matrix .

Use elementary row or column operations to evaluate the determinant.

Find the eigenvalues of the matrix .

Evaluate for the matrix .

Use the fact that for an matrix A, , to evaluate .

The eigenvalues of the matrix are , , . Find a non-zero eigenvector corresponding to each of the given eigenvalues.

Find the eigenvalues of the matrix .

Which property of determinants is illustrated by the equation

Find the minor and its cofactor of the matrix .

Find the determinant of the elementary matrix .

Find the characteristic equation of the matrix .

Find the eigenvalues of the matrix . If an eigenvalue has multiplicity greater than one, list the eigenvalue according to its multiplicity. For example, list an eigenvalue with multiplicity two twice.

Solve for x given the following equation involving a determinant.

Determine whether the points , , and are collinear.

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Matrices

Vector Spaces

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Eigenvalues Eigenvectors

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