Question 1

Find the inverse of the matrix $$\left[ \begin{array} { c c } 
3 & 6 \
- 9 & - 15
\end{array} \right]$$ .

Accepted Answer

A) ​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c } 
3 & 6 \
- 9 & - 15
\end{array} \right]$$ 
B) ​ $$\frac { 1 } { 3 } \left[ \begin{array} { c c } 
- 5 & - 2 \
3 & 1
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
- 3 & - 6 \
15 & 9
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { c c } 
3 & - 9 \
12 & - 21
\end{array} \right]$$ 
E) ​ $$\frac { 1 } { 13 } \left[ \begin{array} { l l } 
- 3 & - 6 \
- 9 & 15
\end{array} \right]$$ 
A) ​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c } 
3 & 6 \
- 9 & - 15
\end{array} \right]$$ 
B) ​ $$\frac { 1 } { 3 } \left[ \begin{array} { c c } 
- 5 & - 2 \
3 & 1
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
- 3 & - 6 \
15 & 9
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { c c } 
3 & - 9 \
12 & - 21
\end{array} \right]$$ 
E) ​ $$\frac { 1 } { 13 } \left[ \begin{array} { l l } 
- 3 & - 6 \
- 9 & 15
\end{array} \right]$$

Question 2

Find the inverse of the matrix.​ $$\left[ \begin{array} { l l l } 
1 & 5 & 2 \
0 & 1 & 5 \
0 & 0 & 1
\end{array} \right]$$ ​

Accepted Answer

A) ​ $$\left[ \begin{array} { c c c } 
1 & 5 & - 23 \
0 & 1 & 5 \
0 & 0 & 1
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c c } 
- 1 & - 5 & 23 \
0 & - 1 & - 5 \
0 & 0 & - 1
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c c } 
1 & - 5 & - 23 \
0 & 1 & - 5 \
0 & 0 & 1
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l l l } 
1 & 5 & 23 \
0 & 1 & 5 \
0 & 0 & 1
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { c c c } 
1 & - 5 & 23 \
0 & 1 & - 5 \
0 & 0 & 1
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { c c c } 
1 & 5 & - 23 \
0 & 1 & 5 \
0 & 0 & 1
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c c } 
- 1 & - 5 & 23 \
0 & - 1 & - 5 \
0 & 0 & - 1
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c c } 
1 & - 5 & - 23 \
0 & 1 & - 5 \
0 & 0 & 1
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l l l } 
1 & 5 & 23 \
0 & 1 & 5 \
0 & 0 & 1
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { c c c } 
1 & - 5 & 23 \
0 & 1 & - 5 \
0 & 0 & 1
\end{array} \right]$$

Question 3

Find the inverse of the matrix $$\left[ \begin{array} { c c } 
- 8 & 14 \
- 4 & 7
\end{array} \right]$$ (if it exists).

Accepted Answer

A) ​ $$\frac { 1 } { 4 } \left[ \begin{array} { l l } 
7 & - 14 \
4 & - 8
\end{array} \right]$$ 
B) ​ $$- \frac { 1 } { 8 } \left[ \begin{array} { c c } 
7 & - 14 \
4 & - 8
\end{array} \right]$$ 
C) ​ $$- \frac { 1 } { 4 } \left[ \begin{array} { c c } 
- 7 & - 14 \
4 & 8
\end{array} \right]$$ 
D) ​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c } 
- 8 & 14 \
- 4 & 7
\end{array} \right]$$ 
E) does not exist 
A) ​ $$\frac { 1 } { 4 } \left[ \begin{array} { l l } 
7 & - 14 \
4 & - 8
\end{array} \right]$$ 
B) ​ $$- \frac { 1 } { 8 } \left[ \begin{array} { c c } 
7 & - 14 \
4 & - 8
\end{array} \right]$$ 
C) ​ $$- \frac { 1 } { 4 } \left[ \begin{array} { c c } 
- 7 & - 14 \
4 & 8
\end{array} \right]$$ 
D) ​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c } 
- 8 & 14 \
- 4 & 7
\end{array} \right]$$ 
E) does not exist

Question 4

​Solve the system of linear equations $$\left\{ \begin{array} { l l } 
- 6 x + 18 y + 6 z & = 1 \
12 x + 30 y & = 2 \
18 x + 6 y - 12 z & = - 1
\end{array} \right.$$ using an inverse matrix. ​

Accepted Answer

A) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 6 } \\
\frac { - 1 } { 2 } \\
\frac { 1 } { 3 }
\end{array} \right]$$ ​ 
B) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 1 } { 6 } \\
\frac { - 1 } { 2 } \\
\frac { - 1 } { 3 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 1 } { 6 } \
0 \
\frac { 1 } { 2 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
0 \
\frac { 1 } { 3 } \
\frac { - 1 } { 6 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 1 } { 6 } \
0 \
\frac { 1 } { 3 }
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 6 } \\
\frac { - 1 } { 2 } \\
\frac { 1 } { 3 }
\end{array} \right]$$ ​ 
B) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 1 } { 6 } \\
\frac { - 1 } { 2 } \\
\frac { - 1 } { 3 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 1 } { 6 } \
0 \
\frac { 1 } { 2 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
0 \
\frac { 1 } { 3 } \
\frac { - 1 } { 6 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 1 } { 6 } \
0 \
\frac { 1 } { 3 }
\end{array} \right]$$

Question 5

Find the inverse of the matrix (if it exists).​ $$\left[ \begin{array} { l l } 
6 & - 7 \
7 & - 8
\end{array} \right]$$ ​

Accepted Answer

A) ​ $$\left[ \begin{array} { c c } 
- 8 & 7 \
7 & 6
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c } 
- 8 & 7 \
- 7 & 6
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
- 8 & - 7 \
- 7 & 6
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { c c } 
- 8 & 7 \
7 & - 6
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { c c } 
- 8 & - 7 \
7 & 6
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { c c } 
- 8 & 7 \
7 & 6
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c } 
- 8 & 7 \
- 7 & 6
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
- 8 & - 7 \
- 7 & 6
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { c c } 
- 8 & 7 \
7 & - 6
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { c c } 
- 8 & - 7 \
7 & 6
\end{array} \right]$$

Question 6

​Solve the system of linear equations $$\left\{ \begin{array} { l } 
3 x + 6 y = 1 \
6 x + 8 y = 2
\end{array} \right.$$ using the inverse matrix $$\left[ \begin{array} { c c } 
- \frac { 2 } { 3 } & \frac { 1 } { 2 } \
\frac { 1 } { 2 } & - \frac { 1 } { 4 }
\end{array} \right]$$

Accepted Answer

A) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 11 } { 12 } \
0
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 1 } { 3 } \
0
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 11 } { 12 } \\
\frac { 1 } { 3 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 3 } \
- \frac { 1 } { 2 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
- \frac { 1 } { 2 } \
\frac { 1 } { 3 }
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 11 } { 12 } \
0
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 1 } { 3 } \
0
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 11 } { 12 } \\
\frac { 1 } { 3 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 3 } \
- \frac { 1 } { 2 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
- \frac { 1 } { 2 } \
\frac { 1 } { 3 }
\end{array} \right]$$

Question 7

Solve the system of linear equations. ​​ $$\left\{ \begin{array} { l } 
x - 2 y = 0 \
2 x - 3 y = 8
\end{array} \right.$$ ​

Accepted Answer

A) ​ $( 16,8 )$ 
B) ​ $( 16 , - 8 )$ 
C) ​ $( - 8,16 )$ 
D) ​ $( 8,16 )$ 
E) ​ $( - 16,8 )$ 
A) ​ $( 16,8 )$ 
B) ​ $( 16 , - 8 )$ 
C) ​ $( - 8,16 )$ 
D) ​ $( 8,16 )$ 
E) ​ $( - 16,8 )$

Question 8

​Use an inverse matrix to solve (if possible)the system of linear equations.​ $$\left\{ \begin{array} { l } 
18 x + 12 y = 14 \
30 x + 24 y = 24
\end{array} \right.$$

Accepted Answer

A) ​( $\frac { 5 } { 6 }$ , $\frac { 1 } { 6 }$ ) 
B) ​( 4, $\frac { 1 } { 6 }$ ) 
C) ​( $\frac { 2 } { 3 }$ , $\frac { 1 } { 6 }$ ) 
D) ​( $\frac { 2 } { 3 }$ , $\frac { 1 } { 4 }$ ) 
E) ​( $\frac { 2 } { 3 }$ ,3 ) 
A) ​( $\frac { 5 } { 6 }$ , $\frac { 1 } { 6 }$ ) 
B) ​( 4, $\frac { 1 } { 6 }$ ) 
C) ​( $\frac { 2 } { 3 }$ , $\frac { 1 } { 6 }$ ) 
D) ​( $\frac { 2 } { 3 }$ , $\frac { 1 } { 4 }$ ) 
E) ​( $\frac { 2 } { 3 }$ ,3 )

Question 9

Find the inverse of the matrix (if it exists).​ $$\left[ \begin{array} { c c } 
- 7 & 33 \
4 & - 19
\end{array} \right]$$ ​

Accepted Answer

A) ​ $$\left[ \begin{array} { c c } 
- 19 & - 33 \
4 & - 7
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c } 
- 19 & - 33 \
- 4 & - 7
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
19 & - 33 \
- 4 & - 7
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { c c } 
19 & - 33 \
4 & - 7
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { c c } 
19 & - 33 \
4 & 7
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { c c } 
- 19 & - 33 \
4 & - 7
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c } 
- 19 & - 33 \
- 4 & - 7
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
19 & - 33 \
- 4 & - 7
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { c c } 
19 & - 33 \
4 & - 7
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { c c } 
19 & - 33 \
4 & 7
\end{array} \right]$$

Question 10

Find the inverse of the matrix (if it exists).​ $$\left[ \begin{array} { c c } 
4 & - 1 \
- 3 & 1
\end{array} \right]$$ ​

Accepted Answer

A) ​ $$\left[ \begin{array} { l l } 
1 & 1 \
3 & 4
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c } 
1 & 1 \
- 3 & 4
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
- 1 & 1 \
3 & 4
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { c c } 
1 & 1 \
3 & - 4
\end{array} \right]$$ 
E) Does not exist 
A) ​ $$\left[ \begin{array} { l l } 
1 & 1 \
3 & 4
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c } 
1 & 1 \
- 3 & 4
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
- 1 & 1 \
3 & 4
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { c c } 
1 & 1 \
3 & - 4
\end{array} \right]$$ 
E) Does not exist

Question 11

​Solve the system of linear equations​ $$\left\{ \begin{array} { l l } 
9 x + 9 y + 9 z & = 1 \
27 x + 45 y + 36 z & = - 3 \
27 x + 54 y + 45 z & = 2
\end{array} \right.$$ ​using the inverse matrix $$\frac { 1 } { 9 } \left[ \begin{array} { c c c } 
1 & 1 & - 1 \
- 3 & 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ .

Accepted Answer

A) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 5 } { 9 } \\
\frac { - 2 } { 3 } \\
\frac { 4 } { 9 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 4 } { 9 } \\
\frac { - 11 } { 9 } \\
\frac { 16 } { 9 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 11 } { 9 } \\\
\frac { 16 } { 9 } \\
\frac { - 4 } { 9 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 5 } { 9 } \\
\frac { 1 } { 3 } \\
\frac { - 2 } { 9 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 16 } { 9 } \\
\frac { - 4 } { 9 } \\
\frac { - 11 } { 9 }
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 5 } { 9 } \\
\frac { - 2 } { 3 } \\
\frac { 4 } { 9 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 4 } { 9 } \\
\frac { - 11 } { 9 } \\
\frac { 16 } { 9 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 11 } { 9 } \\\
\frac { 16 } { 9 } \\
\frac { - 4 } { 9 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 5 } { 9 } \\
\frac { 1 } { 3 } \\
\frac { - 2 } { 9 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 16 } { 9 } \\
\frac { - 4 } { 9 } \\
\frac { - 11 } { 9 }
\end{array} \right]$$

Question 12

Find the inverse of the matrix (if it exists).​ $$\left[ \begin{array} { c c c } 
- 5 & 0 & 0 \
2 & 0 & 0 \
1 & 5 & 7
\end{array} \right]$$ ​

Accepted Answer

A) ​ $$\left[ \begin{array} { l l l } 
5 & 0 & 0 \
2 & 0 & 0 \
1 & 5 & 7
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c c } 
- 5 & 0 & 0 \
- 2 & 0 & 0 \
1 & 5 & 7
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c c } 
- 5 & 0 & 0 \
- 2 & 0 & 0 \
- 1 & - 5 & - 7
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l l l } 
- 5 & 0 & 0 \
- 2 & 0 & 0 \
- 1 & 5 & 7
\end{array} \right]$$ 
E) ​Does not exist 
A) ​ $$\left[ \begin{array} { l l l } 
5 & 0 & 0 \
2 & 0 & 0 \
1 & 5 & 7
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c c } 
- 5 & 0 & 0 \
- 2 & 0 & 0 \
1 & 5 & 7
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c c } 
- 5 & 0 & 0 \
- 2 & 0 & 0 \
- 1 & - 5 & - 7
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l l l } 
- 5 & 0 & 0 \
- 2 & 0 & 0 \
- 1 & 5 & 7
\end{array} \right]$$ 
E) ​Does not exist

Question 13

Use an inverse matrix to solve (if possible)the system of linear equations.​ $$\left\{ \begin{array} { l } 
3 x + 4 y = - 2 \
5 x + 3 y = 4
\end{array} \right.$$ ​

Accepted Answer

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

Question 14

Use the matrix capabilities of a graphing utility to find the inverse of the matrix $$\frac { 1 } { 9 } \left[ \begin{array} { c c c } 
- 4 & - 5 & 3 \
- 4 & - 8 & 3 \
1 & 2 & 0
\end{array} \right]$$ (if it exists).

Accepted Answer

A) ​ $$\left[ \begin{array} { c c c } 
- 9 & 12 & 6 \
0 & - 3 & 0 \
0 & 3 & - 9
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c c } 
- 6 & 6 & 9 \
3 & - 3 & 0 \
0 & 3 & 12
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c c } 
- 6 & 0 & 9 \
0 & - 3 & 0 \
0 & 3 & - 6
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c c } 
3 & 3 & 9 \
0 & - 3 & 0 \
0 & 6 & 12
\end{array} \right]$$ 
E) ​does not exist ​ 
A) ​ $$\left[ \begin{array} { c c c } 
- 9 & 12 & 6 \
0 & - 3 & 0 \
0 & 3 & - 9
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { c c c } 
- 6 & 6 & 9 \
3 & - 3 & 0 \
0 & 3 & 12
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c c } 
- 6 & 0 & 9 \
0 & - 3 & 0 \
0 & 3 & - 6
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c c } 
3 & 3 & 9 \
0 & - 3 & 0 \
0 & 6 & 12
\end{array} \right]$$ 
E) ​does not exist ​

Question 15

Find the inverse of A.​ $$A = \left[ \begin{array} { c c } 
2 & 3 \
- 4 & 1
\end{array} \right]$$ ​

Accepted Answer

A) ​ $$A ^ { - 1 } = \left[ \begin{array} { c c } 
\frac { 1 } { 7 } & - \frac { 3 } { 7 } \\
\frac { 2 } { 7 } & \frac { 1 } { 7 }
\end{array} \right]$$ 
B) ​ $$A ^ { - 1 } = \left[ \begin{array} { c c } 
\frac { 1 } { 14 } & - \frac { 3 } { 14 } \\
\frac { 2 } { 7 } & \frac { 1 } { 7 }
\end{array} \right]$$ 
C) ​ $$A ^ { - 1 } = \left[ \begin{array} { c c } 
\frac { 1 } { 14 } & - \frac { 3 } { 7 } \\
\frac { 2 } { 7 } & \frac { 1 } { 14 }
\end{array} \right]$$ 
D) ​ $$A ^ { - 1 } = \left[ \begin{array} { c c } 
\frac { 1 } { 14 } & \frac { 3 } { 14 } \
\frac { 2 } { 7 } & \frac { 1 } { 7 }
\end{array} \right]$$ 
E) ​ $$A ^ { - 1 } = \left[ \begin{array} { c c } 
\frac { 1 } { 7 } & - \frac { 3 } { 7 } \
\frac { 2 } { 14 } & \frac { 1 } { 14 }
\end{array} \right]$$ 
A) ​ $$A ^ { - 1 } = \left[ \begin{array} { c c } 
\frac { 1 } { 7 } & - \frac { 3 } { 7 } \\
\frac { 2 } { 7 } & \frac { 1 } { 7 }
\end{array} \right]$$ 
B) ​ $$A ^ { - 1 } = \left[ \begin{array} { c c } 
\frac { 1 } { 14 } & - \frac { 3 } { 14 } \\
\frac { 2 } { 7 } & \frac { 1 } { 7 }
\end{array} \right]$$ 
C) ​ $$A ^ { - 1 } = \left[ \begin{array} { c c } 
\frac { 1 } { 14 } & - \frac { 3 } { 7 } \\
\frac { 2 } { 7 } & \frac { 1 } { 14 }
\end{array} \right]$$ 
D) ​ $$A ^ { - 1 } = \left[ \begin{array} { c c } 
\frac { 1 } { 14 } & \frac { 3 } { 14 } \
\frac { 2 } { 7 } & \frac { 1 } { 7 }
\end{array} \right]$$ 
E) ​ $$A ^ { - 1 } = \left[ \begin{array} { c c } 
\frac { 1 } { 7 } & - \frac { 3 } { 7 } \
\frac { 2 } { 14 } & \frac { 1 } { 14 }
\end{array} \right]$$

Question 16

​Solve the system of linear equations $$\left\{ \begin{array} { l } 
2 x + 3 y = - 15 \
4 x - 3 y = 10
\end{array} \right.$$ using the inverse matrix $$\left[ \begin{array} { c c } 
\frac { 1 } { 6 } & \frac { 1 } { 6 } \\
\frac { 2 } { 9 } & - \frac { 1 } { 9 }
\end{array} \right]$$

Accepted Answer

A) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 5 } { 6 } \\
\frac { 5 } { 2 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
- \frac { 5 } { 6 } \\
- \frac { 40 } { 9 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 25 } { 18 } \\
\frac { 5 } { 2 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 5 } { 2 } \
0
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
- \frac { 5 } { 3 } \\
\frac { 5 } { 6 }
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 5 } { 6 } \\
\frac { 5 } { 2 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
- \frac { 5 } { 6 } \\
- \frac { 40 } { 9 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 25 } { 18 } \\
\frac { 5 } { 2 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 5 } { 2 } \
0
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
- \frac { 5 } { 3 } \\
\frac { 5 } { 6 }
\end{array} \right]$$

Question 17

Consider a person who invests in AAA-rated bonds,A-rated bonds,and B-rated bonds.The average yields are 6.5% on AAA bonds,7% on A bonds,and 9% on B bonds.The person invests twice as much in B bonds as in A bonds.Let x,y and z represent the amounts invested in AAA,A,and B bonds,respectively. $$\begin{array} { | c | c | } 
\hline \text { Total Investment } & \text { Annual Return } \
\hline \$ 12,000 & 890 \
\hline
\end{array}$$ $$\left\{ \begin{array} { c l } 
x + y + z & = 663,000 \text { (total investment) } \
0.065 x + 0.07 y + 0.09 z & = 47,000 \text { (annual retum) } \
2 y - z & = 0
\end{array} \right.$$ Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond.
​

Accepted Answer

A) $142,000 in AAA-rated bonds $451,000 in A-rated bonds
$71,000 in B-rated bonds 
B) $450,000 in AAA-rated bonds $71,000 in A-rated bonds
$142,000 in B-rated bonds 
C) $143,000 in AAA-rated bonds $71,000 in A-rated bonds
$450,000 in B-rated bonds 
D) $72,000 in AAA-rated bonds $450,000 in A-rated bonds
$142,000 in B-rated bonds 
E) $450,000 in AAA-rated bonds $142,000 in A-rated bonds
$70,000 in B-rated bonds 
A) $142,000 in AAA-rated bonds $451,000 in A-rated bonds
$71,000 in B-rated bonds 
B) $450,000 in AAA-rated bonds $71,000 in A-rated bonds
$142,000 in B-rated bonds 
C) $143,000 in AAA-rated bonds $71,000 in A-rated bonds
$450,000 in B-rated bonds 
D) $72,000 in AAA-rated bonds $450,000 in A-rated bonds
$142,000 in B-rated bonds 
E) $450,000 in AAA-rated bonds $142,000 in A-rated bonds
$70,000 in B-rated bonds

Question 18

​Solve the system of linear equations $$\left\{ \begin{array} { l l } 
5 x _ { 1 } - 10 x _ { 2 } - 5 x _ { 3 } - 10 x _ { 4 } & = 0 \
15 x _ { 1 } - 25 x _ { 2 } - 10 x _ { 3 } - 15 x _ { 4 } & = 8 \
10 x _ { 1 } - 25 x _ { 2 } - 10 x _ { 3 } - 25 x _ { 4 } & = - 8 \
- 5 x _ { 1 } + 20 x _ { 2 } + 20 x _ { 3 } + 55 x _ { 4 } & = 16
\end{array} \right.$$ using the inverse matrix $$\frac { 1 } { 5 } \left[ \begin{array} { c c c c } 
- 24 & 7 & 1 & - 2 \
- 10 & 3 & 0 & - 1 \
- 29 & 7 & 3 & - 2 \
12 & - 3 & - 1 & 1
\end{array} \right]$$ .

Accepted Answer

A) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 16 } { 5 } \\
\frac { 8 } { 5 } \\
\frac { - 8 } { 5 } \\
\frac { - 16 } { 5 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
0 \\
\frac { 8 } { 5 } \\
0 \\
\frac { 16 } { 5 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 16 } { 5 } \\
\frac { 8 } { 5 } \\
0 \\
0
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 24 } { 5 } \\
\frac { - 8 } { 5 } \\
0 \\
\frac { - 16 } { 5 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 24 } { 5 } \\
\frac { - 8 } { 5 } \\
\frac { - 16 } { 5 } \\
\frac { 16 } { 5 }
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 16 } { 5 } \\
\frac { 8 } { 5 } \\
\frac { - 8 } { 5 } \\
\frac { - 16 } { 5 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
0 \\
\frac { 8 } { 5 } \\
0 \\
\frac { 16 } { 5 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 16 } { 5 } \\
\frac { 8 } { 5 } \\
0 \\
0
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 24 } { 5 } \\
\frac { - 8 } { 5 } \\
0 \\
\frac { - 16 } { 5 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 24 } { 5 } \\
\frac { - 8 } { 5 } \\
\frac { - 16 } { 5 } \\
\frac { 16 } { 5 }
\end{array} \right]$$

Question 19

Find the inverse of the matrix $$\left[ \begin{array} { c c c } 
8 & 8 & 8 \
24 & 40 & 32 \
24 & 48 & 40
\end{array} \right]$$ .

Accepted Answer

A) ​ $$\frac { 1 } { 8 } \left[ \begin{array} { c c c } 
1 & 1 & - 1 \
- 3 & 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ 
B) ​ $$\frac { 1 } { 8 } \left[ \begin{array} { c c c } 
1 & 1 & 1 \
0 & - 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ 
C) ​ $$8 \left[ \begin{array} { c c c } 
1 & 1 & 1 \
- 3 & 0 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ 
D) ​ $$- \frac { 1 } { 8 } \left[ \begin{array} { c c c } 
1 & 0 & - 1 \
- 3 & 2 & - 1 \
2 & - 3 & 3
\end{array} \right]$$ 
E) ​ $$- 8 \left[ \begin{array} { c c c } 
3 & 1 & 0 \
- 3 & 1 & - 1 \
1 & - 3 & 2
\end{array} \right]$$ 
A) ​ $$\frac { 1 } { 8 } \left[ \begin{array} { c c c } 
1 & 1 & - 1 \
- 3 & 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ 
B) ​ $$\frac { 1 } { 8 } \left[ \begin{array} { c c c } 
1 & 1 & 1 \
0 & - 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ 
C) ​ $$8 \left[ \begin{array} { c c c } 
1 & 1 & 1 \
- 3 & 0 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ 
D) ​ $$- \frac { 1 } { 8 } \left[ \begin{array} { c c c } 
1 & 0 & - 1 \
- 3 & 2 & - 1 \
2 & - 3 & 3
\end{array} \right]$$ 
E) ​ $$- 8 \left[ \begin{array} { c c c } 
3 & 1 & 0 \
- 3 & 1 & - 1 \
1 & - 3 & 2
\end{array} \right]$$

Question 20

​Use the matrix capabilities of a graphing utility to solve the following system of linear equations: $$\left\{ \begin{array} { c c c } 
15 x - 5 y & = & 3 \
10 x + 10 y & = & 6 \
20 z & = & - 12
\end{array} \right.$$

Accepted Answer

A) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 3 } { 10 } \\
\frac { 9 } { 10 } \\
\frac { - 3 } { 5 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 10 } \
0 \
\frac { - 3 } { 5 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 3 } { 10 } \\
\frac { 3 } { 10 } \\
\frac { - 3 } { 5 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 3 } { 10 } \\
\frac { - 3 } { 10 } \\
0
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 3 } { 10 } \\
\frac { 3 } { 10 } \\
\frac { - 3 } { 5 }
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 3 } { 10 } \\
\frac { 9 } { 10 } \\
\frac { - 3 } { 5 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 10 } \
0 \
\frac { - 3 } { 5 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 3 } { 10 } \\
\frac { 3 } { 10 } \\
\frac { - 3 } { 5 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 3 } { 10 } \\
\frac { - 3 } { 10 } \\
0
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 3 } { 10 } \\
\frac { 3 } { 10 } \\
\frac { - 3 } { 5 }
\end{array} \right]$$

Exam 49: The Inverse of a Square Matrix

Find the inverse of the matrix [36−9−15]\left[ \begin{array} { c c } 3 & 6 \\- 9 & - 15\end{array} \right][3−9​6−15​] .

Find the inverse of the matrix.​ [152015001]\left[ \begin{array} { l l l } 1 & 5 & 2 \\0 & 1 & 5 \\0 & 0 & 1\end{array} \right]​100​510​251​​ ​

Find the inverse of the matrix [−814−47]\left[ \begin{array} { c c } - 8 & 14 \\- 4 & 7\end{array} \right][−8−4​147​] (if it exists).

​Solve the system of linear equations {−6x+18y+6z=112x+30y=218x+6y−12z=−1\left\{ \begin{array} { l l } - 6 x + 18 y + 6 z & = 1 \\12 x + 30 y & = 2 \\18 x + 6 y - 12 z & = - 1\end{array} \right.⎩⎨⎧​−6x+18y+6z12x+30y18x+6y−12z​=1=2=−1​ using an inverse matrix. ​

Find the inverse of the matrix (if it exists).​ [6−77−8]\left[ \begin{array} { l l } 6 & - 7 \\7 & - 8\end{array} \right][67​−7−8​] ​

Solve the system of linear equations. ​​ {x−2y=02x−3y=8\left\{ \begin{array} { l } x - 2 y = 0 \\2 x - 3 y = 8\end{array} \right.{x−2y=02x−3y=8​ ​

​Use an inverse matrix to solve (if possible)the system of linear equations.​ {18x+12y=1430x+24y=24\left\{ \begin{array} { l } 18 x + 12 y = 14 \\30 x + 24 y = 24\end{array} \right.{18x+12y=1430x+24y=24​

Find the inverse of the matrix (if it exists).​ [−7334−19]\left[ \begin{array} { c c } - 7 & 33 \\4 & - 19\end{array} \right][−74​33−19​] ​

Find the inverse of the matrix (if it exists).​ [4−1−31]\left[ \begin{array} { c c } 4 & - 1 \\- 3 & 1\end{array} \right][4−3​−11​] ​

Find the inverse of the matrix (if it exists).​ [−500200157]\left[ \begin{array} { c c c } - 5 & 0 & 0 \\2 & 0 & 0 \\1 & 5 & 7\end{array} \right]​−521​005​007​​ ​

Use an inverse matrix to solve (if possible)the system of linear equations.​ {3x+4y=−25x+3y=4\left\{ \begin{array} { l } 3 x + 4 y = - 2 \\5 x + 3 y = 4\end{array} \right.{3x+4y=−25x+3y=4​ ​

Use the matrix capabilities of a graphing utility to find the inverse of the matrix 19[−4−53−4−83120]\frac { 1 } { 9 } \left[ \begin{array} { c c c } - 4 & - 5 & 3 \\- 4 & - 8 & 3 \\1 & 2 & 0\end{array} \right]91​​−4−41​−5−82​330​​ (if it exists).

Find the inverse of A.​ A=[23−41]A = \left[ \begin{array} { c c } 2 & 3 \\- 4 & 1\end{array} \right]A=[2−4​31​] ​

Find the inverse of the matrix [888244032244840]\left[ \begin{array} { c c c } 8 & 8 & 8 \\24 & 40 & 32 \\24 & 48 & 40\end{array} \right]​82424​84048​83240​​ .

​Use the matrix capabilities of a graphing utility to solve the following system of linear equations: {15x−5y=310x+10y=620z=−12\left\{ \begin{array} { c c c } 15 x - 5 y & = & 3 \\10 x + 10 y & = & 6 \\20 z & = & - 12\end{array} \right.⎩⎨⎧​15x−5y10x+10y20z​===​36−12​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses

Hyperbolas

Rotation of Conics

Parametric Equations

Polar Coordinates

Find the inverse of the matrix $\left[ \begin{array} { c c } 3 & 6 \\- 9 & - 15\end{array} \right]$ .

Find the inverse of the matrix. $\left[ \begin{array} { l l l } 1 & 5 & 2 \\0 & 1 & 5 \\0 & 0 & 1\end{array} \right]$

Find the inverse of the matrix $\left[ \begin{array} { c c } - 8 & 14 \\- 4 & 7\end{array} \right]$ (if it exists).

Solve the system of linear equations $\left\{ \begin{array} { l l } - 6 x + 18 y + 6 z & = 1 \\12 x + 30 y & = 2 \\18 x + 6 y - 12 z & = - 1\end{array} \right.$ using an inverse matrix.

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { l l } 6 & - 7 \\7 & - 8\end{array} \right]$

Solve the system of linear equations. $\left\{ \begin{array} { l } x - 2 y = 0 \\2 x - 3 y = 8\end{array} \right.$

Use an inverse matrix to solve (if possible)the system of linear equations. $\left\{ \begin{array} { l } 18 x + 12 y = 14 \\30 x + 24 y = 24\end{array} \right.$

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { c c } - 7 & 33 \\4 & - 19\end{array} \right]$

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { c c } 4 & - 1 \\- 3 & 1\end{array} \right]$

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { c c c } - 5 & 0 & 0 \\2 & 0 & 0 \\1 & 5 & 7\end{array} \right]$

Use an inverse matrix to solve (if possible)the system of linear equations. $\left\{ \begin{array} { l } 3 x + 4 y = - 2 \\5 x + 3 y = 4\end{array} \right.$

Use the matrix capabilities of a graphing utility to find the inverse of the matrix $\frac { 1 } { 9 } \left[ \begin{array} { c c c } - 4 & - 5 & 3 \\- 4 & - 8 & 3 \\1 & 2 & 0\end{array} \right]$ (if it exists).

Find the inverse of A. $A = \left[ \begin{array} { c c } 2 & 3 \\- 4 & 1\end{array} \right]$

Find the inverse of the matrix $\left[ \begin{array} { c c c } 8 & 8 & 8 \\24 & 40 & 32 \\24 & 48 & 40\end{array} \right]$ .

Use the matrix capabilities of a graphing utility to solve the following system of linear equations: $\left\{ \begin{array} { c c c } 15 x - 5 y & = & 3 \\10 x + 10 y & = & 6 \\20 z & = & - 12\end{array} \right.$