Question 1

Find the inverse of the matrix A, if it exists.
-$$A = \left[ \begin{array} { r r r } 
8 & - 4 & 2 \
11 & - 7 & 4 \
3 & - 3 & 2
\end{array} \right]$$

Accepted Answer

A)  $A ^ { - 1 }$ does not exist. 
B)  $$\text {  } A ^ { - 1 } = \left[ \begin{array} { c c c } 
\frac { 1 } { 8 } & \frac { 1 } { 11 } & \frac { 1 } { 2 } \
\frac { 1 } { 11 } & - \frac { 1 } { 7 } & \frac { 1 } { 4 } \
\frac { 1 } { 3 } & - \frac { 1 } { 3 } & \frac { 1 } { 2 }
\end{array} \right]$$ 
C)  $A ^ { - 1 } = \left[ \begin{array} { c c c } \frac { 2 } { 11 } & \frac { 2 } { 11 } & - 2 \ \frac { 3 } { 11 } & \frac { 8 } { 7 } & 2 \ \frac { 8 } { 3 } & - \frac { 2 } { 3 } & \frac { 1 } { 2 } \end{array} \right]$ 
D)  $\mathrm { A } ^ { - 1 } = \left[ \begin{array} { r r r } 8 & 11 & 3 \ - 4 & - 7 & - 3 \ 2 & 4 & 2 \end{array} \right]$ 
A)  $A ^ { - 1 }$ does not exist. 
B)  $$\text {  } A ^ { - 1 } = \left[ \begin{array} { c c c } 
\frac { 1 } { 8 } & \frac { 1 } { 11 } & \frac { 1 } { 2 } \
\frac { 1 } { 11 } & - \frac { 1 } { 7 } & \frac { 1 } { 4 } \
\frac { 1 } { 3 } & - \frac { 1 } { 3 } & \frac { 1 } { 2 }
\end{array} \right]$$ 
C)  $A ^ { - 1 } = \left[ \begin{array} { c c c } \frac { 2 } { 11 } & \frac { 2 } { 11 } & - 2 \ \frac { 3 } { 11 } & \frac { 8 } { 7 } & 2 \ \frac { 8 } { 3 } & - \frac { 2 } { 3 } & \frac { 1 } { 2 } \end{array} \right]$ 
D)  $\mathrm { A } ^ { - 1 } = \left[ \begin{array} { r r r } 8 & 11 & 3 \ - 4 & - 7 & - 3 \ 2 & 4 & 2 \end{array} \right]$

Question 2

Decide whether or not the matrices are inverses of each other.
-$\left[ \begin{array} { r r r } 2 & - 1 & 0 \ - 1 & 1 & - 2 \ 1 & 0 & - 1 \end{array} \right]$ and $\left[ \begin{array} { r r r } 1 & - 1 & 2 \ - 3 & - 2 & 4 \ - 1 & 1 & 1 \end{array} \right]$

Accepted Answer

A)  $Y e s$ 
B)  $\mathrm { No }$ 
A)  $Y e s$ 
B)  $\mathrm { No }$

Question 3

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
- 5 x _ { 1 } + 3 x _ { 2 } = 8 \
- 2 x _ { 1 } + 4 x _ { 2 } = 20
\end{array}$$

Accepted Answer

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

Question 4

Find the matrix product AB, if it is defined.
-$$A = \left[ \begin{array} { r r } 
- 1 & 3 \
1 & 6
\end{array} \right] , B = \left[ \begin{array} { l l l } 
0 & - 2 & 6 \
1 & - 3 & 2
\end{array} \right]$$

Accepted Answer

A)  $\mathrm { AB }$ is undefined. 
B) 
$$\left[ \begin{array} { r r } 
0 & - 6 \
18 & 1 \
- 18 & 12
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r } 
3 & 6 & - 7 \
- 20 & 0 & 18
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r r } 
3 & - 7 & 0 \
6 & - 20 & 18
\end{array} \right]$$ 
A)  $\mathrm { AB }$ is undefined. 
B) 
$$\left[ \begin{array} { r r } 
0 & - 6 \
18 & 1 \
- 18 & 12
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r } 
3 & 6 & - 7 \
- 20 & 0 & 18
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r r } 
3 & - 7 & 0 \
6 & - 20 & 18
\end{array} \right]$$

Question 5

Find the matrix product AB, if it is defined.
-$$A = \left[ \begin{array} { r r } 
0 & - 2 \
4 & 3
\end{array} \right] , B = \left[ \begin{array} { r r r } 
- 1 & 3 & 2 \
0 & - 3 & 1
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r } 
0 & - 6 & - 8 \
0 & - 9 & 3
\end{array} \right]$$ 
B)

$$\left[ \begin{array} { r r r } 
0 & 6 & - 2 \
- 4 & 3 & 11
\end{array} \right]$$ 
C)  $\mathrm { AB }$ is undefined. 
D) 
$$\left[ \begin{array} { r r r } 
0 & - 4 & 6 \
3 & - 2 & 11
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r r } 
0 & - 6 & - 8 \
0 & - 9 & 3
\end{array} \right]$$ 
B)

$$\left[ \begin{array} { r r r } 
0 & 6 & - 2 \
- 4 & 3 & 11
\end{array} \right]$$ 
C)  $\mathrm { AB }$ is undefined. 
D) 
$$\left[ \begin{array} { r r r } 
0 & - 4 & 6 \
3 & - 2 & 11
\end{array} \right]$$

Question 6

Identify the indicated submatrix.
-$A = \left[ \begin{array} { r r | r } 2 & 6 & 1 \ - 2 & 0 & - 1 \ 0 & 3 & - 6 \ \hline 3 & 6 & 3 \end{array} \right] .$

Accepted Answer

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

Question 7

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

Accepted Answer

A) 
$\left[ \begin{array} { r r } - 7 & 4 \ 4 & 4 \ 10 & - 4 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } 11 & - 12 \ - 8 & - 11 \ - 4 & 1 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 11 & - 12 \ 8 & - 5 \ - 4 & - 1 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 11 & - 5 \ - 8 & - 11 \ - 4 & 1 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r } - 7 & 4 \ 4 & 4 \ 10 & - 4 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } 11 & - 12 \ - 8 & - 11 \ - 4 & 1 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 11 & - 12 \ 8 & - 5 \ - 4 & - 1 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 11 & - 5 \ - 8 & - 11 \ - 4 & 1 \end{array} \right]$

Question 8

Find a basis for the null space of the matrix.
-$$A = \left[ \begin{array} { r r r r r } 
1 & 0 & - 4 & 0 & - 4 \
0 & 1 & 2 & 0 & 2 \
0 & 0 & 0 & 1 & 1 \
0 & 0 & 0 & 0 & 0
\end{array} \right]$$

Accepted Answer

A) 
$ \left\{\left[\begin{array}{r}-4 \ 2 \ 1 \ 0 \ 0\end{array}\right],\left[\begin{array}{r}-4 \ 2 \ 0 \ -1 \ 1\end{array}\right]\right\} $ 
B) 
$$\left\{ \left[ \begin{array} { r } 
- 4 \
2 \
1 \
0 \
0
\end{array} \right] , \left[ \begin{array} { r } 
- 4 \
2 \
0 \
- 1 \
1
\end{array} \right] \right\}$$ 
C) 
$\left\{\left[\begin{array}{r}
1 \
0 \
-4 \
0 \
-4
\end{array}\right],\left[\begin{array}{l}
0 \
1 \
2 \
0 \
2
\end{array}\right]\right\}$ 
D) 
$\left\{\left[\begin{array}{l}
1 \
0 \
0 \
0
\end{array}\right],\left[\begin{array}{l}
0 \
1 \
0 \
0
\end{array}\right],\left[\begin{array}{l}
0 \
0 \
1 \
0
\end{array}\right]\right\}$ 
A) 
$ \left\{\left[\begin{array}{r}-4 \ 2 \ 1 \ 0 \ 0\end{array}\right],\left[\begin{array}{r}-4 \ 2 \ 0 \ -1 \ 1\end{array}\right]\right\} $ 
B) 
$$\left\{ \left[ \begin{array} { r } 
- 4 \
2 \
1 \
0 \
0
\end{array} \right] , \left[ \begin{array} { r } 
- 4 \
2 \
0 \
- 1 \
1
\end{array} \right] \right\}$$ 
C) 
$\left\{\left[\begin{array}{r}
1 \
0 \
-4 \
0 \
-4
\end{array}\right],\left[\begin{array}{l}
0 \
1 \
2 \
0 \
2
\end{array}\right]\right\}$ 
D) 
$\left\{\left[\begin{array}{l}
1 \
0 \
0 \
0
\end{array}\right],\left[\begin{array}{l}
0 \
1 \
0 \
0
\end{array}\right],\left[\begin{array}{l}
0 \
0 \
1 \
0
\end{array}\right]\right\}$

Question 9

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are
defined.
-$A$ is $2 \times 4$, B is $2 \times 4$.

Accepted Answer

A)  $\mathrm { AB }$ is $2 \times 4 , \mathrm { BA }$ is $4 \times 2$. 
B)  $\mathrm { AB }$ is $2 \times 2 , \mathrm { BA }$ is $4 \times 4$. 
C)  $A B$ is undefined, $B A$ is undefined. 
D)  $\mathrm { AB }$ is $4 \times 2$, $\mathrm { BA }$ is $2 \times 4$. 
A)  $\mathrm { AB }$ is $2 \times 4 , \mathrm { BA }$ is $4 \times 2$. 
B)  $\mathrm { AB }$ is $2 \times 2 , \mathrm { BA }$ is $4 \times 4$. 
C)  $A B$ is undefined, $B A$ is undefined. 
D)  $\mathrm { AB }$ is $4 \times 2$, $\mathrm { BA }$ is $2 \times 4$.

Question 10

Find a basis for the null space of the matrix.
-$$A = \left[ \begin{array} { r r r r } 
1 & 0 & - 7 & - 4 \
0 & 1 & 5 & - 2 \
0 & 0 & 0 & 0
\end{array} \right]$$

Accepted Answer

A) $$\left\{ \left[ \begin{array} { l } 
1 \
0 \
0
\end{array} \right] , \left[ \begin{array} { l } 
0 \
1 \
0
\end{array} \right] \right\}$$ 
B)

$$\left\{ \left[ \begin{array} { r } 
- 7 \
5 \
1 \
0
\end{array} \right] , \left[ \begin{array} { r } 
- 4 \
- 2 \
0 \
1
\end{array} \right] \right\}$$ 
C) 
$\left\{\left[\begin{array}{r}
7 \
-5 \
1 \
0
\end{array}\right],\left[\begin{array}{l}
4 \
2 \
0 \
1
\end{array}\right]\right\}$ 
D)

$\left\{\left[\begin{array}{r}
1 \
0 \
-7 \
-4
\end{array}\right],\left[\begin{array}{r}
0 \
1 \
5 \
-2
\end{array}\right]\right\}$ 
A) $$\left\{ \left[ \begin{array} { l } 
1 \
0 \
0
\end{array} \right] , \left[ \begin{array} { l } 
0 \
1 \
0
\end{array} \right] \right\}$$ 
B)

$$\left\{ \left[ \begin{array} { r } 
- 7 \
5 \
1 \
0
\end{array} \right] , \left[ \begin{array} { r } 
- 4 \
- 2 \
0 \
1
\end{array} \right] \right\}$$ 
C) 
$\left\{\left[\begin{array}{r}
7 \
-5 \
1 \
0
\end{array}\right],\left[\begin{array}{l}
4 \
2 \
0 \
1
\end{array}\right]\right\}$ 
D)

$\left\{\left[\begin{array}{r}
1 \
0 \
-7 \
-4
\end{array}\right],\left[\begin{array}{r}
0 \
1 \
5 \
-2
\end{array}\right]\right\}$

Question 11

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are
defined.
-A is $4 \times 4$, B is $4 \times 4$.

Accepted Answer

A)  $\mathrm { AB }$ is $8 \times 4$, BA is $8 \times 4$. 
B)  $\mathrm { AB }$ is $4 \times 4$, BA is $4 \times 4$. 
C)  $\mathrm { AB }$ is $4 \times 8$, BA is $4 \times 8$. 
D)  $\mathrm { AB }$ is $1 \times 1$, BA is $1 \times 1$. 
A)  $\mathrm { AB }$ is $8 \times 4$, BA is $8 \times 4$. 
B)  $\mathrm { AB }$ is $4 \times 4$, BA is $4 \times 4$. 
C)  $\mathrm { AB }$ is $4 \times 8$, BA is $4 \times 8$. 
D)  $\mathrm { AB }$ is $1 \times 1$, BA is $1 \times 1$.

Question 12

Determine the rank of the matrix.
-$$\left[ \begin{array} { r r r r r } 
1 & 0 & - 4 & 0 & 4 \
0 & 1 & - 3 & 0 & 4 \
0 & 0 & 0 & 1 & 1 \
0 & 0 & 0 & 0 & 0
\end{array} \right]$$

Accepted Answer

A) 5 
B) 3 
C) 4 
D) 2 
A) 5 
B) 3 
C) 4 
D) 2

Question 13

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { c } 
- 3 x _ { 1 } - 2 x _ { 2 } = 2 \
6 x _ { 1 } + 4 x _ { 2 } = 8
\end{array}$$

Accepted Answer

A)  $( - 2 , - 2 )$ 
B)  $\left( - \frac { 2 } { 3 } + \frac { 3 } { 2 } x _ { 2 } , x _ { 2 } \right)$ 
C)  $( 2,8 )$ 
D)  No solution 
A)  $( - 2 , - 2 )$ 
B)  $\left( - \frac { 2 } { 3 } + \frac { 3 } { 2 } x _ { 2 } , x _ { 2 } \right)$ 
C)  $( 2,8 )$ 
D)  No solution

Question 14

Find the transpose of the matrix.
-$\left[ \begin{array} { r r } 8 & 4 \ - 4 & 0 \ - 7 & 7 \end{array} \right]$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r } 
4 & 0 & 7 \
8 & - 4 & - 7
\end{array} \right]$$ 
B) 
$\left[\begin{array}{rrr}
8 & -4 & -7 \
4 & 0 & 7
\end{array}\right]$ 
C) 
$\left[\begin{array}{rr}
-7 & 7 \
-4 & 0 \
8 & 4
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr}
4 & 8 \
0 & -4 \
7 & -7
\end{array}\right]$ 
A) 
$$\left[ \begin{array} { r r r } 
4 & 0 & 7 \
8 & - 4 & - 7
\end{array} \right]$$ 
B) 
$\left[\begin{array}{rrr}
8 & -4 & -7 \
4 & 0 & 7
\end{array}\right]$ 
C) 
$\left[\begin{array}{rr}
-7 & 7 \
-4 & 0 \
8 & 4
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr}
4 & 8 \
0 & -4 \
7 & -7
\end{array}\right]$

Question 15

Find the matrix product AB for the partitioned matrices.
-$$A = \left[ \begin{array} { r r | r } 
4 & 0 & 1 \
2 & - 1 & - 3 \
5 & 3 & 7
\end{array} \right] , B = \left[ \begin{array} { r r r | r } 
- 2 & 0 & 8 & 5 \
1 & 6 & 2 & 2 \
\hline 4 & - 1 & 0 & 3
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r | r } 
- 4 & - 1 & 32 & 23 \
- 17 & - 3 & 14 & - 1 \
21 & 11 & 46 & 52
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r | r } 
- 4 & - 1 & 0 & 3 \
- 12 & - 3 & 0 & - 9 \
\hline 28 & - 7 & 0 & 21
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r | r } 
- 8 & 0 & 32 & 20 \
- 5 & - 6 & 14 & 8 \
- 7 & 18 & 46 & 31
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { r r r | r } 
- 4 & - 1 & 32 & 23 \
- 17 & - 3 & 14 & - 1 \
\hline 21 & 11 & 46 & 52
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r r | r } 
- 4 & - 1 & 32 & 23 \
- 17 & - 3 & 14 & - 1 \
21 & 11 & 46 & 52
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r | r } 
- 4 & - 1 & 0 & 3 \
- 12 & - 3 & 0 & - 9 \
\hline 28 & - 7 & 0 & 21
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r | r } 
- 8 & 0 & 32 & 20 \
- 5 & - 6 & 14 & 8 \
- 7 & 18 & 46 & 31
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { r r r | r } 
- 4 & - 1 & 32 & 23 \
- 17 & - 3 & 14 & - 1 \
\hline 21 & 11 & 46 & 52
\end{array} \right]$$

Question 16

Perform the matrix operation.
-Let $C = \left[ \begin{array} { r } 1 \ - 3 \ 2 \end{array} \right]$ and $D = \left[ \begin{array} { r } - 1 \ 3 \ - 2 \end{array} \right]$. Find $C - 2 D$

Accepted Answer

A) 
$\left[ \begin{array} { r } 3 \ - 9 \ 6 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r } - 1 \ 3 \ - 2 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r } 3 \ - 6 \ 4 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r } - 3 \ 9 \ - 6 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r } 3 \ - 9 \ 6 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r } - 1 \ 3 \ - 2 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r } 3 \ - 6 \ 4 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r } - 3 \ 9 \ - 6 \end{array} \right]$

Question 17

Find the matrix product AB for the partitioned matrices.
-$$A = \left[ \begin{array} { l l } 
0 & I \
I & F
\end{array} \right] , B = \left[ \begin{array} { l l } 
W & X \
Y & Z
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { l l } X & W + X F \ Z & Y + Z F \end{array} \right]$ 
B) 
$$\left[ \begin{array} { l l } 
\mathrm { Y } & Z \
W + Y F & X + Z F
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { l l } 
0 & Z \
F Y & F Z
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { l l } 
Y & Z \
W + F Y & X + F Z
\end{array} \right]$$ 
A)  $\left[ \begin{array} { l l } X & W + X F \ Z & Y + Z F \end{array} \right]$ 
B) 
$$\left[ \begin{array} { l l } 
\mathrm { Y } & Z \
W + Y F & X + Z F
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { l l } 
0 & Z \
F Y & F Z
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { l l } 
Y & Z \
W + F Y & X + F Z
\end{array} \right]$$

Question 18

Find the 3 × 3 matrix that produces the described composite 2D transformation, using homogeneous coordinates.
-Rotate points through 45° and then scale the x-coordinate by 0.6 and the y-coordinate by 0.8.

Accepted Answer

A) 
$\left[ \begin{array} { c c c } 0.3 & - 0.4 \sqrt { 2 } & 0 \ 0.3 \sqrt { 2 } & 0.4 & 0 \ 0 & 0 & 1 \end{array} \right]$ 
B)

$\left[ \begin{array} { c c c } 0.3 \sqrt { 2 } & 0.3 \sqrt { 2 } & 0 \ - 0.4 \sqrt { 2 } & 0.4 \sqrt { 2 } & 0 \ 0 & 0 & 1 \end{array} \right]$ 
C) 
$$\left[ \begin{array} { c c c } 
0.3 \sqrt { 2 } & - 0.3 \sqrt { 2 } & 0 \
0.4 \sqrt { 2 } & 0.4 \sqrt { 2 } & 0 \
0 & 0 & 1
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c c } 
0 & - 0.6 & 0 \
0.8 & 0 & 0 \
0 & 0 & 1
\end{array} \right]$$ 
A) 
$\left[ \begin{array} { c c c } 0.3 & - 0.4 \sqrt { 2 } & 0 \ 0.3 \sqrt { 2 } & 0.4 & 0 \ 0 & 0 & 1 \end{array} \right]$ 
B)

$\left[ \begin{array} { c c c } 0.3 \sqrt { 2 } & 0.3 \sqrt { 2 } & 0 \ - 0.4 \sqrt { 2 } & 0.4 \sqrt { 2 } & 0 \ 0 & 0 & 1 \end{array} \right]$ 
C) 
$$\left[ \begin{array} { c c c } 
0.3 \sqrt { 2 } & - 0.3 \sqrt { 2 } & 0 \
0.4 \sqrt { 2 } & 0.4 \sqrt { 2 } & 0 \
0 & 0 & 1
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c c } 
0 & - 0.6 & 0 \
0.8 & 0 & 0 \
0 & 0 & 1
\end{array} \right]$$

Question 19

Determine the production vector x that will satisfy demand in an economy with the given consumption matrix C and
final demand vector d. Round production levels to the nearest whole number.
-$$\mathrm { C } = \left[ \begin{array} { l l l } 
.2 & .1 & .1 \
.3 & .2 & .3 \
.4 & .1 & .3
\end{array} \right] , \mathrm { d } = \left[ \begin{array} { l } 
213 \
323 \
298
\end{array} \right]$$

Accepted Answer

A)  $\mathbf { x } = \left[ \begin{array} { r } 728 \ 978 \ - 302 \end{array} \right]$ 
B)  $\mathbf { x } = \left[ \begin{array} { r } 108 \ 105 \ 91 \end{array} \right]$ 
C)  $\mathbf { x } = \left[ \begin{array} { l } 105 \ 218 \ 207 \end{array} \right]$ 
D)  $\mathbf { x } = \left[ \begin{array} { l } 482 \ 895 \ 829 \end{array} \right]$ 
A)  $\mathbf { x } = \left[ \begin{array} { r } 728 \ 978 \ - 302 \end{array} \right]$ 
B)  $\mathbf { x } = \left[ \begin{array} { r } 108 \ 105 \ 91 \end{array} \right]$ 
C)  $\mathbf { x } = \left[ \begin{array} { l } 105 \ 218 \ 207 \end{array} \right]$ 
D)  $\mathbf { x } = \left[ \begin{array} { l } 482 \ 895 \ 829 \end{array} \right]$

Question 20

Find the inverse of the matrix, if it exists
-$A = \left[ \begin{array} { r r } - 5 & - 5 \ 2 & 2 \end{array} \right]$

Accepted Answer

A)  A is not invertible 
B) 
$$\left[ \begin{array} { c c } 
\frac { 2 } { 21 } & \frac { 5 } { 21 } \
- \frac { 2 } { 21 } & - \frac { 5 } { 21 }
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c c } 
- \frac { 2 } { 21 } & - \frac { 5 } { 21 } \
\frac { 2 } { 21 } & \frac { 5 } { 21 }
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { l } 
\frac { 2 } { 21 } - \frac { 5 } { 21 } \
\frac { 2 } { 21 } - \frac { 5 } { 21 }
\end{array} \right]$$ 
A)  A is not invertible 
B) 
$$\left[ \begin{array} { c c } 
\frac { 2 } { 21 } & \frac { 5 } { 21 } \
- \frac { 2 } { 21 } & - \frac { 5 } { 21 }
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c c } 
- \frac { 2 } { 21 } & - \frac { 5 } { 21 } \
\frac { 2 } { 21 } & \frac { 5 } { 21 }
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { l } 
\frac { 2 } { 21 } - \frac { 5 } { 21 } \
\frac { 2 } { 21 } - \frac { 5 } { 21 }
\end{array} \right]$$

Find the inverse of the matrix A, if it exists. - $A = \left[ \begin{array} { r r r } 8 & - 4 & 2 \\11 & - 7 & 4 \\3 & - 3 & 2\end{array} \right]$

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { r r r } 2 & - 1 & 0 \\ - 1 & 1 & - 2 \\ 1 & 0 & - 1 \end{array} \right]$ and $\left[ \begin{array} { r r r } 1 & - 1 & 2 \\ - 3 & - 2 & 4 \\ - 1 & 1 & 1 \end{array} \right]$

Solve the system by using the inverse of the coefficient matrix. - -5+3=8 -2+4=20

Find the matrix product AB, if it is defined. - $A = \left[ \begin{array} { r r } - 1 & 3 \\1 & 6\end{array} \right] , B = \left[ \begin{array} { l l l } 0 & - 2 & 6 \\1 & - 3 & 2\end{array} \right]$

Find the matrix product AB, if it is defined. - $A = \left[ \begin{array} { r r } 0 & - 2 \\4 & 3\end{array} \right] , B = \left[ \begin{array} { r r r } - 1 & 3 & 2 \\0 & - 3 & 1\end{array} \right]$

Identify the indicated submatrix. - $A = \left[ \begin{array} { r r | r } 2 & 6 & 1 \\ - 2 & 0 & - 1 \\ 0 & 3 & - 6 \\ \hline 3 & 6 & 3 \end{array} \right] .$

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

Find a basis for the null space of the matrix. - $A = \left[ \begin{array} { r r r r r } 1 & 0 & - 4 & 0 & - 4 \\0 & 1 & 2 & 0 & 2 \\0 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 0 & 0\end{array} \right]$

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined. - $A$ is $2 \times 4$ , B is $2 \times 4$ .

Find a basis for the null space of the matrix. - $A = \left[ \begin{array} { r r r r } 1 & 0 & - 7 & - 4 \\0 & 1 & 5 & - 2 \\0 & 0 & 0 & 0\end{array} \right]$

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined. -A is $4 \times 4$ , B is $4 \times 4$ .

Determine the rank of the matrix. - $\left[ \begin{array} { r r r r r } 1 & 0 & - 4 & 0 & 4 \\0 & 1 & - 3 & 0 & 4 \\0 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 0 & 0\end{array} \right]$

Solve the system by using the inverse of the coefficient matrix. - -3-2=2 6+4=8

Find the transpose of the matrix. - $\left[ \begin{array} { r r } 8 & 4 \\ - 4 & 0 \\ - 7 & 7 \end{array} \right]$

Find the matrix product AB for the partitioned matrices. - $A = \left[ \begin{array} { r r | r } 4 & 0 & 1 \\2 & - 1 & - 3 \\5 & 3 & 7\end{array} \right] , B = \left[ \begin{array} { r r r | r } - 2 & 0 & 8 & 5 \\1 & 6 & 2 & 2 \\\hline 4 & - 1 & 0 & 3\end{array} \right]$

Perform the matrix operation. -Let $C = \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]$ and $D = \left[ \begin{array} { r } - 1 \\ 3 \\ - 2 \end{array} \right]$ . Find $C - 2 D$

Find the matrix product AB for the partitioned matrices. - $A = \left[ \begin{array} { l l } 0 & I \\I & F\end{array} \right] , B = \left[ \begin{array} { l l } W & X \\Y & Z\end{array} \right]$

Find the 3 × 3 matrix that produces the described composite 2D transformation, using homogeneous coordinates. -Rotate points through 45° and then scale the x-coordinate by 0.6 and the y-coordinate by 0.8.

Find the inverse of the matrix, if it exists - $A = \left[ \begin{array} { r r } - 5 & - 5 \\ 2 & 2 \end{array} \right]$

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Exam 2: Matrix Algebra

Find the inverse of the matrix A, if it exists. - A=[8−4211−743−32]A = \left[ \begin{array} { r r r } 8 & - 4 & 2 \\11 & - 7 & 4 \\3 & - 3 & 2\end{array} \right]A=​8113​−4−7−3​242​​

Solve the system by using the inverse of the coefficient matrix. - -5+3=8 -2+4=20

Find the matrix product AB, if it is defined. - A=[−1316],B=[0−261−32]A = \left[ \begin{array} { r r } - 1 & 3 \\1 & 6\end{array} \right] , B = \left[ \begin{array} { l l l } 0 & - 2 & 6 \\1 & - 3 & 2\end{array} \right]A=[−11​36​],B=[01​−2−3​62​]

Find the matrix product AB, if it is defined. - A=[0−243],B=[−1320−31]A = \left[ \begin{array} { r r } 0 & - 2 \\4 & 3\end{array} \right] , B = \left[ \begin{array} { r r r } - 1 & 3 & 2 \\0 & - 3 & 1\end{array} \right]A=[04​−23​],B=[−10​3−3​21​]

Identify the indicated submatrix. - A=[261−20−103−6363].A = \left[ \begin{array} { r r | r } 2 & 6 & 1 \\ - 2 & 0 & - 1 \\ 0 & 3 & - 6 \\ \hline 3 & 6 & 3 \end{array} \right] .A=​2−203​6036​1−1−63​​​.

Find a basis for the null space of the matrix. - A=[10−40−4012020001100000]A = \left[ \begin{array} { r r r r r } 1 & 0 & - 4 & 0 & - 4 \\0 & 1 & 2 & 0 & 2 \\0 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 0 & 0\end{array} \right]A=​1000​0100​−4200​0010​−4210​​

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined. - AAA is 2×42 \times 42×4 , B is 2×42 \times 42×4 .

Find a basis for the null space of the matrix. - A=[10−7−4015−20000]A = \left[ \begin{array} { r r r r } 1 & 0 & - 7 & - 4 \\0 & 1 & 5 & - 2 \\0 & 0 & 0 & 0\end{array} \right]A=​100​010​−750​−4−20​​

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined. -A is 4×44 \times 44×4 , B is 4×44 \times 44×4 .

Determine the rank of the matrix. - [10−40401−3040001100000]\left[ \begin{array} { r r r r r } 1 & 0 & - 4 & 0 & 4 \\0 & 1 & - 3 & 0 & 4 \\0 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 0 & 0\end{array} \right]​1000​0100​−4−300​0010​4410​​