Question 1

Solve for x given the following equation involving a determinant.

Accepted Answer

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

Question 2

Use the matrix capabilities of a graphing utility to find the determinant of the matrix   .

Accepted Answer

A)  -162 
B)  0 
C)  -216 
D)  -188 
E)  188 
A)  -162 
B)  0 
C)  -216 
D)  -188 
E)  188 E

Question 3

Evaluate   for the matrix   .

Accepted Answer

A)  8 
B)  784 
C)  -220 
D)  4 
E)  484 
A)  8 
B)  784 
C)  -220 
D)  4 
E)  484 D

Question 4

Find the determinant of   by the method of expansion by cofactors.

Accepted Answer

A)  192 
B)  384 
C)  -48 
D)  -384 
E)  176 
A)  192 
B)  384 
C)  -48 
D)  -384 
E)  176

Question 5

Use technology to find the determinant.

Accepted Answer

A)  119 
B)  -119 
C)  -207 
D)  -115 
E)  121 
A)  119 
B)  -119 
C)  -207 
D)  -115 
E)  121

Question 6

Find the characteristic equation of the matrix   .

Accepted Answer

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

Question 7

Determine whether the points   ,   ,   , and   are coplanar.

Accepted Answer

A) True 
 B)False

Question 8

The eigenvalues of the matrix   are   and   . Find an eigenvector corresponding to each of the given eigenvalues.

Accepted Answer

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

Question 9

Find the minor   and its cofactor   of the matrix   .

Accepted Answer

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

Question 10

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 11

Use elementary row or column operations to evaluate the determinant.

Accepted Answer

A)  160 
B)  0 
C)  -160 
D)  120 
E)  -120 
A)  160 
B)  0 
C)  -160 
D)  120 
E)  -120

Question 12

Which property of determinants is illustrated by the equation

Accepted Answer

A)  If every entry of an   matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c. 
B)  If a single row or a single column of an   matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c. 
C)  If a single row or a single column of an   matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by   . 
D)  If every entry of an   matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by   
E)  If c times one row is added to another row of an   matrix, the determinant of the matrix is multiplied by c. 
A)  If every entry of an   matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c. 
B)  If a single row or a single column of an   matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by c. 
C)  If a single row or a single column of an   matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by   . 
D)  If every entry of an   matrix is multiplied by a constant c, then the determinant of the matrix is multiplied by   
E)  If c times one row is added to another row of an   matrix, the determinant of the matrix is multiplied by c.

Question 13

Use Cramer's Rule to solve the following system of linear equations   .

Accepted Answer

A)  x =    , y =   , z =   
B) x =   , y =   , z =    
C)  x =   , y =   , z =    
D)  x =   , y =    , z =   
E)  x =   , y =   , z =   
A)  x =    , y =   , z =   
B) x =   , y =   , z =    
C)  x =   , y =   , z =    
D)  x =   , y =    , z =   
E)  x =   , y =   , z =

Question 14

Find the volume of the tetrahedron having the vertices   ,   ,   , and   .

Accepted Answer

A)  2 
B)  12 
C)  26 
D)  4 
E)  31 
A)  2 
B)  12 
C)  26 
D)  4 
E)  31

Question 15

Find   ,   and   .   ,   .

Accepted Answer

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

Question 16

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

Accepted Answer

A)  -420 
B)  -1,050 
C)  -26,250 
D)  1,050 
E)  26,250 
A)  -420 
B)  -1,050 
C)  -26,250 
D)  1,050 
E)  26,250

Question 17

Evaluate the determinant of the matrix below to determine which of the following makes the equation true.

Accepted Answer

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

Question 18

Find the area of the triangle having the vertices   ,   , and   .

Accepted Answer

A)  12 
B)  6 
C)  30 
D)  69 
E)  9 
A)  12 
B)  6 
C)  30 
D)  69 
E)  9

Question 19

Use Cramer's Rule to solve the system of equations   .

Accepted Answer

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

Question 20

Find the determinant of the elementary matrix   .

Accepted Answer

A)  1 
B)  1- k 
C)  k 
D)  -k 
E)  k - 1 
A)  1 
B)  1- k 
C)  k 
D)  -k 
E)  k - 1

Solve for x given the following equation involving a determinant.

Use the matrix capabilities of a graphing utility to find the determinant of the matrix .

Evaluate for the matrix .

Find the determinant of by the method of expansion by cofactors.

Use technology to find the determinant.

Find the characteristic equation of the matrix .

Determine whether the points , , , and are coplanar.

The eigenvalues of the matrix are and . Find an eigenvector corresponding to each of the given eigenvalues.

Find the minor and its cofactor of the matrix .

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

Use elementary row or column operations to evaluate the determinant.

Which property of determinants is illustrated by the equation

Use Cramer's Rule to solve the following system of linear equations .

Find the volume of the tetrahedron having the vertices , , , and .

Find , and . , .

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

Evaluate the determinant of the matrix below to determine which of the following makes the equation true.

Find the area of the triangle having the vertices , , and .

Use Cramer's Rule to solve the system of equations .

Find the determinant of the elementary matrix .

Linear Equations

Matrices

Vector Spaces

Inner Product Spaces

Linear Transformations

Eigenvalues Eigenvectors

Filters

Exam 3: Determinants

Solve for x given the following equation involving a determinant.

Use the matrix capabilities of a graphing utility to find the determinant of the matrix .

Evaluate for the matrix .

Find the determinant of by the method of expansion by cofactors.

Use technology to find the determinant.

Find the characteristic equation of the matrix .

Determine whether the points , , , and are coplanar.

The eigenvalues of the matrix are and . Find an eigenvector corresponding to each of the given eigenvalues.

Find the minor and its cofactor of the matrix .

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

Use elementary row or column operations to evaluate the determinant.

Which property of determinants is illustrated by the equation

Use Cramer's Rule to solve the following system of linear equations .

Find the volume of the tetrahedron having the vertices , , , and .

Find , and . , .

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

Evaluate the determinant of the matrix below to determine which of the following makes the equation true.

Find the area of the triangle having the vertices , , and .

Use Cramer's Rule to solve the system of equations .

Find the determinant of the elementary matrix .

Linear Equations

Matrices

Vector Spaces

Inner Product Spaces

Linear Transformations

Eigenvalues Eigenvectors

Filters