Question 1

Let (m × n) denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
-(4 × 4)(4 × 4)

Accepted Answer

A) 4 × 4 
B) 8 × 4 
C) not defined 
D) 4 × 8 
A) 4 × 4 
B) 8 × 4 
C) not defined 
D) 4 × 8

Question 2

Find the indicated matrix.
-Let $A = \left[ \begin{array} { r r r } 7 & - 4 & 8 \ - 6 & 5 & - 1 \ 0 & 6 & - 3 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } - 2 & - 6 & - 1 \ - 7 & - 4 & 3 \ - 3 & - 9 & - 5 \end{array} \right]$. Find $A - B$

Accepted Answer

A)  $\left[ \begin{array} { r r r } 5 & - 10 & 7 \ - 13 & 1 & 2 \ - 3 & - 3 & - 8 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r r } 9 & 2 & 9 \ 1 & 9 & - 4 \ 3 & 15 & 2 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r r } 5 & - 10 & 7 \ - 13 & 1 & - 8 \ - 3 & - 3 & 2 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } 9 & 2 & 9 \ 1 & 9 & 2 \ 3 & 15 & - 4 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r r } 5 & - 10 & 7 \ - 13 & 1 & 2 \ - 3 & - 3 & - 8 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r r } 9 & 2 & 9 \ 1 & 9 & - 4 \ 3 & 15 & 2 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r r } 5 & - 10 & 7 \ - 13 & 1 & - 8 \ - 3 & - 3 & 2 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } 9 & 2 & 9 \ 1 & 9 & 2 \ 3 & 15 & - 4 \end{array} \right]$

Question 3

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these.
-$$\mathrm { A } = \left[ \begin{array} { r r } 
3 & - 4 \
5 & 0 \
- 5 & 9 \
4 & - 3
\end{array} \right]$$

Accepted Answer

A) 4 × 2; none of these 
B) 2; square matrix 
C) 8; none of these 
D) 2 × 4; none of these 
A) 4 × 2; none of these 
B) 2; square matrix 
C) 8; none of these 
D) 2 × 4; none of these

Question 4

Find the indicated matrix.
-Let $\mathrm { A } = \left[ \begin{array} { l l } 2 & 3 \ 2 & 4 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 0 & 4 \ - 1 & 6 \end{array} \right]$. Find $4 \mathrm {~A} + \mathrm { B }$.

Accepted Answer

A)  $\left[ \begin{array} { l l } 8 & 16 \ 1 & 10 \end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 8 & 7 \ 7 & 10 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 8 & 16 \ 7 & 22 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l } 8 & 28 \ 4 & 40 \end{array} \right]$ 
A)  $\left[ \begin{array} { l l } 8 & 16 \ 1 & 10 \end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 8 & 7 \ 7 & 10 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 8 & 16 \ 7 & 22 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l } 8 & 28 \ 4 & 40 \end{array} \right]$

Question 5

Determine whether matrix B is the inverse of matrix A by finding the product AB.
-$$A = \left[ \begin{array} { r r } 
- 5 & - 1 \
6 & 0
\end{array} \right] , B = \left[ \begin{array} { c c } 
0 & \frac { 1 } { 6 } \
- 1 & \frac { 5 } { 6 }
\end{array} \right]$$

Accepted Answer

A)  $B \neq A ^ { - 1 }$ 
B)  $B = A ^ { - 1 }$ 
A)  $B \neq A ^ { - 1 }$ 
B)  $B = A ^ { - 1 }$

Question 6

Solve the linear system using an inverse matrix.
-$$\begin{aligned}
x + 2 y + 3 z & = 1 \
x + y + z & = 10 \
x - 2 z & = 7
\end{aligned}$$

Accepted Answer

A)  $( 31 , - 33,12 )$ 
B)  $( 21 , - 32,12 )$ 
C)  $( 49,67,28 )$ 
D)  $( 1,0,0 )$ 
A)  $( 31 , - 33,12 )$ 
B)  $( 21 , - 32,12 )$ 
C)  $( 49,67,28 )$ 
D)  $( 1,0,0 )$

Question 7

Find the indicated matrix.
-Let $\mathrm { A } = \left[ \begin{array} { c c } - 1 & - 1 \ - 8 & - 2 \ 8 & - 1 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 4 & 8 \ - 4 & 1 \ - 1 & - 1 \end{array} \right]$. Find $\mathrm { A } + \mathrm { B }$.

Accepted Answer

A)  $\left[ \begin{array} { r r } - 5 & 7 \ - 12 & - 1 \ 7 & - 2 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 5 & - 2 \ - 12 & - 1 \ 7 & - 2 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 3 & - 9 \ - 4 & - 3 \ 9 & - 9 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 5 & 7 \ 12 & - 2 \ 7 & 2 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r } - 5 & 7 \ - 12 & - 1 \ 7 & - 2 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 5 & - 2 \ - 12 & - 1 \ 7 & - 2 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 3 & - 9 \ - 4 & - 3 \ 9 & - 9 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 5 & 7 \ 12 & - 2 \ 7 & 2 \end{array} \right]$

Question 8

Find the indicated matrix.
-Let $\mathrm { A } = \left[ \begin{array} { r r } 2 & - 1 \ 7 & 9 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 5 & - 3 \ 4 & 7 \end{array} \right]$. Find $- 2 \mathrm {~A} + 4 \mathrm {~B}$.

Accepted Answer

A)  $\left[ \begin{array} { l r } 7 & 4 \ 11 & 13 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l } - 24 & - 18 \ - 30 & - 34 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 16 & - 10 \ 2 & 10 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 3 & - 6 \ 3 & 5 \end{array} \right]$ 
A)  $\left[ \begin{array} { l r } 7 & 4 \ 11 & 13 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l } - 24 & - 18 \ - 30 & - 34 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 16 & - 10 \ 2 & 10 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 3 & - 6 \ 3 & 5 \end{array} \right]$

Question 9

Use Cramer's rule to solve the linear system.
-$$\begin{array} { l } 
2 x + 2 y = 2 \
3 x + y = - 3
\end{array}$$

Accepted Answer

A)  $( 2 , - 3 )$ 
B)  $( 3 , - 2 )$ 
C)  $( - 2,3 )$ 
D)  $( - 3 , - 2 )$ 
A)  $( 2 , - 3 )$ 
B)  $( 3 , - 2 )$ 
C)  $( - 2,3 )$ 
D)  $( - 3 , - 2 )$

Question 10

Let (m × n) denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
-(2 × 1)(1 × 1)

Accepted Answer

A) 1 × 2 
B) 2 × 1 
C) not defined 
D) 2 × 2 
A) 1 × 2 
B) 2 × 1 
C) not defined 
D) 2 × 2

Question 11

Use Cramer's rule to solve the linear system.
-$$\begin{array} { l } 
3 x + 5 y = 34 \
4 x + 5 y = 37
\end{array}$$

Accepted Answer

A)  $( 3,5 )$ 
B)  $( - 3 , - 5 )$ 
C)  $( 5,3 )$ 
D)  $( - 5,3 )$ 
A)  $( 3,5 )$ 
B)  $( - 3 , - 5 )$ 
C)  $( 5,3 )$ 
D)  $( - 5,3 )$

Question 12

Solve the linear system using an inverse matrix.
-$$\begin{aligned}
- 5 x + 3 y & = 8 \
3 x - 6 y & = - 30
\end{aligned}$$

Accepted Answer

A)  $( - 6 , - 2 )$ 
B)  $( 6,2 )$ 
C)  $( - 2 , - 6 )$ 
D)  $( 2,6 )$ 
A)  $( - 6 , - 2 )$ 
B)  $( 6,2 )$ 
C)  $( - 2 , - 6 )$ 
D)  $( 2,6 )$

Question 13

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these.
-$$A = \left[ \begin{array} { r r r } 
2 & 0 & 4 \
- 2 & 5 & - 5 \
0 & 8 & - 7
\end{array} \right]$$

Accepted Answer

A) 3 × 3; square matrix 
B) 9; none of these 
C) 6 × 6; square matrix 
D) 2 × 3; none of these 
A) 3 × 3; square matrix 
B) 9; none of these 
C) 6 × 6; square matrix 
D) 2 × 3; none of these

Question 14

Find the inverse of the invertible matrix.
-$$\left[ \begin{array} { l l } 
- 4 & - 4 \
- 3 & - 6
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c } \frac { 1 } { 4 } - \frac { 1 } { 3 } \ - \frac { 1 } { 2 } \end{array} \right.$ 
B)  $\left[ \begin{array} { c c } - \frac { 1 } { 2 } & \frac { 1 } { 3 } \ \frac { 1 } { 4 } & - \frac { 1 } { 3 } \end{array} \right]$ 
C)  $\left[ \begin{array} { c c } - \frac { 1 } { 3 } & \frac { 1 } { 3 } \ \frac { 1 } { 4 } & - \frac { 1 } { 2 } \end{array} \right]$ 
D)  $\left[ \begin{array} { l l } - \frac { 1 } { 2 } & - \frac { 1 } { 3 } \ - \frac { 1 } { 4 } & - \frac { 1 } { 3 } \end{array} \right]$ 
A)  $\left[ \begin{array} { c } \frac { 1 } { 4 } - \frac { 1 } { 3 } \ - \frac { 1 } { 2 } \end{array} \right.$ 
B)  $\left[ \begin{array} { c c } - \frac { 1 } { 2 } & \frac { 1 } { 3 } \ \frac { 1 } { 4 } & - \frac { 1 } { 3 } \end{array} \right]$ 
C)  $\left[ \begin{array} { c c } - \frac { 1 } { 3 } & \frac { 1 } { 3 } \ \frac { 1 } { 4 } & - \frac { 1 } { 2 } \end{array} \right]$ 
D)  $\left[ \begin{array} { l l } - \frac { 1 } { 2 } & - \frac { 1 } { 3 } \ - \frac { 1 } { 4 } & - \frac { 1 } { 3 } \end{array} \right]$

Question 15

Solve the linear system using an inverse matrix.
-$\begin{aligned} - 2 x - 6 y & = - 2 \ 2 x - y & = - 5 \end{aligned}$

Accepted Answer

A)  $( 2 , - 1 )$ 
B)  $( - 2,1 )$ 
C)  $( - 1,2 )$ 
D)  $( 1 , - 2 )$ 
A)  $( 2 , - 1 )$ 
B)  $( - 2,1 )$ 
C)  $( - 1,2 )$ 
D)  $( 1 , - 2 )$

Question 16

Determine whether matrix B is the inverse of matrix A by finding the product AB.
-$$A = \left[ \begin{array} { l l } 
5 & 3 \
3 & 2
\end{array} \right] , B = \left[ \begin{array} { r r } 
2 & - 3 \
- 3 & 5
\end{array} \right]$$

Accepted Answer

A)  $\mathrm { B } = \mathrm { A } ^ { - 1 }$ 
B)  $B \neq A ^ { - 1 }$ 
A)  $\mathrm { B } = \mathrm { A } ^ { - 1 }$ 
B)  $B \neq A ^ { - 1 }$

Question 17

Solve the linear system using an inverse matrix.
-$$\begin{array} { l } 
- 3 x + 9 y = 9 \
3 x + 2 y = 13
\end{array}$$

Accepted Answer

A)  $( 3,2 )$ 
B)  $( - 3 , - 2 )$ 
C)  $( 2,3 )$ 
D)  $( - 2 , - 3 )$ 
A)  $( 3,2 )$ 
B)  $( - 3 , - 2 )$ 
C)  $( 2,3 )$ 
D)  $( - 2 , - 3 )$

Question 18

Let (m × n) denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
-(3 × 2)(2 × 3)

Accepted Answer

A) not defined 
B) 3 × 3 
C) 2 × 2 
D) 2 × 3 
A) not defined 
B) 3 × 3 
C) 2 × 2 
D) 2 × 3

Question 19

Find the inverse of the invertible matrix.
-$$\left[ \begin{array} { c c } 
0 & 1 \
- 6 & - 4
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c c } - \frac { 2 } { 3 } & - \frac { 1 } { 6 } \ 1 & 0 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } 0 & - \frac { 1 } { 6 } \ 1 & - \frac { 2 } { 3 } \end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 1 & 0 \ - \frac { 2 } { 3 } & - \frac { 1 } { 6 } \end{array} \right]$ 
D)  $\left[ \begin{array} { c c } - \frac { 2 } { 3 } & \frac { 1 } { 6 } \ - 1 & 0 \end{array} \right]$ 
A)  $\left[ \begin{array} { c c } - \frac { 2 } { 3 } & - \frac { 1 } { 6 } \ 1 & 0 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } 0 & - \frac { 1 } { 6 } \ 1 & - \frac { 2 } { 3 } \end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 1 & 0 \ - \frac { 2 } { 3 } & - \frac { 1 } { 6 } \end{array} \right]$ 
D)  $\left[ \begin{array} { c c } - \frac { 2 } { 3 } & \frac { 1 } { 6 } \ - 1 & 0 \end{array} \right]$

Question 20

Find the indicated matrix.
-Let $\mathrm { A } = \left[ \begin{array} { r r r } - 4 & 6 & 7 \ 3 & - 5 & 12 \ 7 & - 11 & 14 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } 6 & 10 & - 4 \ - 5 & 6 & - 8 \ 3 & 11 & 7 \end{array} \right]$. Find $\mathrm { A } - \mathrm { B }$

Accepted Answer

A)  $\left[ \begin{array} { c c c } 10 & 4 & - 11 \ - 8 & 11 & - 20 \ - 4 & 22 & - 7 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r r } 2 & 16 & 3 \ - 2 & 1 & 4 \ 10 & 0 & 21 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r r } - 10 & - 4 & 11 \ 8 & - 11 & 20 \ 4 & - 22 & 7 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } - 10 & - 4 & 3 \ - 2 & - 11 & 4 \ 4 & - 22 & 7 \end{array} \right]$ 
A)  $\left[ \begin{array} { c c c } 10 & 4 & - 11 \ - 8 & 11 & - 20 \ - 4 & 22 & - 7 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r r } 2 & 16 & 3 \ - 2 & 1 & 4 \ 10 & 0 & 21 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r r } - 10 & - 4 & 11 \ 8 & - 11 & 20 \ 4 & - 22 & 7 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } - 10 & - 4 & 3 \ - 2 & - 11 & 4 \ 4 & - 22 & 7 \end{array} \right]$

Let (m × n) denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined. -(4 × 4)(4 × 4)

Find the indicated matrix. -Let $A = \left[ \begin{array} { r r r } 7 & - 4 & 8 \\ - 6 & 5 & - 1 \\ 0 & 6 & - 3 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } - 2 & - 6 & - 1 \\ - 7 & - 4 & 3 \\ - 3 & - 9 & - 5 \end{array} \right]$ . Find $A - B$

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these. - $\mathrm { A } = \left[ \begin{array} { r r } 3 & - 4 \\5 & 0 \\- 5 & 9 \\4 & - 3\end{array} \right]$

Find the indicated matrix. -Let $\mathrm { A } = \left[ \begin{array} { l l } 2 & 3 \\ 2 & 4 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 0 & 4 \\ - 1 & 6 \end{array} \right]$ . Find $4 \mathrm {~A} + \mathrm { B }$ .

Determine whether matrix B is the inverse of matrix A by finding the product AB. - $A = \left[ \begin{array} { r r } - 5 & - 1 \\6 & 0\end{array} \right] , B = \left[ \begin{array} { c c } 0 & \frac { 1 } { 6 } \\- 1 & \frac { 5 } { 6 }\end{array} \right]$

Solve the linear system using an inverse matrix. - x+2y+3z =1 x+y+z =10 x-2z =7

Find the indicated matrix. -Let $\mathrm { A } = \left[ \begin{array} { c c } - 1 & - 1 \\ - 8 & - 2 \\ 8 & - 1 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 4 & 8 \\ - 4 & 1 \\ - 1 & - 1 \end{array} \right]$ . Find $\mathrm { A } + \mathrm { B }$ .

Find the indicated matrix. -Let $\mathrm { A } = \left[ \begin{array} { r r } 2 & - 1 \\ 7 & 9 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 5 & - 3 \\ 4 & 7 \end{array} \right]$ . Find $- 2 \mathrm {~A} + 4 \mathrm {~B}$ .

Use Cramer's rule to solve the linear system. - 2x+2y=2 3x+y=-3

Let (m × n) denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined. -(2 × 1)(1 × 1)

Use Cramer's rule to solve the linear system. - 3x+5y=34 4x+5y=37

Solve the linear system using an inverse matrix. - -5x+3y =8 3x-6y =-30

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these. - $A = \left[ \begin{array} { r r r } 2 & 0 & 4 \\- 2 & 5 & - 5 \\0 & 8 & - 7\end{array} \right]$

Find the inverse of the invertible matrix. - $\left[ \begin{array} { l l } - 4 & - 4 \\- 3 & - 6\end{array} \right]$

Solve the linear system using an inverse matrix. - -2x-6y =-2 2x-y =-5

Determine whether matrix B is the inverse of matrix A by finding the product AB. - $A = \left[ \begin{array} { l l } 5 & 3 \\3 & 2\end{array} \right] , B = \left[ \begin{array} { r r } 2 & - 3 \\- 3 & 5\end{array} \right]$

Solve the linear system using an inverse matrix. - -3x+9y=9 3x+2y=13

Let (m × n) denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined. -(3 × 2)(2 × 3)

Find the inverse of the invertible matrix. - $\left[ \begin{array} { c c } 0 & 1 \\- 6 & - 4\end{array} \right]$

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Exam 8: Matrices