Question 1

Solve the system of linear equations.​ $$\left\{ \begin{array} { l } 
x - 2 y = 3 \
2 x - 3 y = - 4
\end{array} \right.$$ ​

Accepted Answer

A) ​ $( - 17,4 )$ 
B) ​ $( - 17,10 )$ 
C) ​ $( - 4 , - 17 )$ 
D) ​ $( 17,10 )$ 
E) ​ $( - 17 , - 10 )$ 
A) ​ $( - 17,4 )$ 
B) ​ $( - 17,10 )$ 
C) ​ $( - 4 , - 17 )$ 
D) ​ $( 17,10 )$ 
E) ​ $( - 17 , - 10 )$ E

Question 2

​Use an inverse matrix to solve (if possible)the system of linear equations.​ $$\left\{ \begin{array} { l l } 
1.8 x - 5 y & = 4 \
28.8 x - 16 y & = 5
\end{array} \right.$$ ​

Accepted Answer

A) ​( $\frac { 1 } { 16 }$ ,16) 
B) ​(0,0) 
C) ​( $\frac { 1 } { 16 }$ , $\frac { 5 } { 16 }$ ) 
D) ​(16,16) 
E) ​No solution 
A) ​( $\frac { 1 } { 16 }$ ,16) 
B) ​(0,0) 
C) ​( $\frac { 1 } { 16 }$ , $\frac { 5 } { 16 }$ ) 
D) ​(16,16) 
E) ​No solution E

Question 3

Find the inverse of A.​ $$A = \left[ \begin{array} { c c c } 
- 4 & 12 & 4 \
8 & 20 & 0 \
12 & 4 & - 8
\end{array} \right]$$ ​

Accepted Answer

A) ​ $$A ^ { - 1 } = \frac { 1 } { 36 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$$ 
B) ​ $$A ^ { - 1 } = \frac { 1 } { 24 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & 1 & 2 \
13 & 10 & 11
\end{array} \right]$$ 
C) ​ $$A ^ { - 1 } = \frac { 1 } { 28 } \left[ \begin{array} { c c c } 
- 10 & 7 & - 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$$ 
D) ​ $$A ^ { - 1 } = \frac { 1 } { 36 } \left[ \begin{array} { c c c } 
- 10 & 7 & - 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$$ 
E) ​ $$A ^ { - 1 } = \frac { 1 } { 28 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & 1 & 2 \
13 & 10 & 11
\end{array} \right]$$ 
A) ​ $$A ^ { - 1 } = \frac { 1 } { 36 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$$ 
B) ​ $$A ^ { - 1 } = \frac { 1 } { 24 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & 1 & 2 \
13 & 10 & 11
\end{array} \right]$$ 
C) ​ $$A ^ { - 1 } = \frac { 1 } { 28 } \left[ \begin{array} { c c c } 
- 10 & 7 & - 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$$ 
D) ​ $$A ^ { - 1 } = \frac { 1 } { 36 } \left[ \begin{array} { c c c } 
- 10 & 7 & - 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$$ 
E) ​ $$A ^ { - 1 } = \frac { 1 } { 28 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & 1 & 2 \
13 & 10 & 11
\end{array} \right]$$ D

Question 4

​Solve the system of linear equations $$\left\{ \begin{array} { l l } 
4 x _ { 1 } - 8 x _ { 2 } - 4 x _ { 3 } - 8 x _ { 4 } & = 0 \
12 x _ { 1 } - 20 x _ { 2 } - 8 x _ { 3 } - 12 x _ { 4 } & = - 15 \
8 x _ { 1 } - 20 x _ { 2 } - 8 x _ { 3 } - 20 x _ { 4 } & = 10 \
- 4 x _ { 1 } + 16 x _ { 2 } + 16 x _ { 3 } + 44 x _ { 4 } & = 0
\end{array} \right.$$ using the inverse matrix $$\frac { 1 } { 4 } \left[ \begin{array} { c c c c } 
- 24 & 7 & 1 & - 2 \
- 10 & 3 & 0 & - 1 \
- 29 & 7 & 3 & - 2 \
12 & - 3 & - 1 & 1
\end{array} \right]$$ .

Accepted Answer

A) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 95 } { 4 } \\
\frac { - 45 } { 4 } \\
\frac { - 75 } { 4 } \\
\frac { 35 } { 4 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
0 \
\frac { - 5 } { 4 } \
0 \
\frac { - 5 } { 2 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 45 } { 4 } \\
\frac { - 95 } { 4 } \\
0 \\
\frac { 35 } { 4 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { r } 
\frac { - 15 } { 4 } \
\frac { 5 } { 4 } \
0 \
\frac { 5 } { 2 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 35 } { 4 } \\
\frac { - 25 } { 4 } \\
\frac { 45 } { 4 } \\
\frac { - 45 } { 4 }
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 95 } { 4 } \\
\frac { - 45 } { 4 } \\
\frac { - 75 } { 4 } \\
\frac { 35 } { 4 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
0 \
\frac { - 5 } { 4 } \
0 \
\frac { - 5 } { 2 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 45 } { 4 } \\
\frac { - 95 } { 4 } \\
0 \\
\frac { 35 } { 4 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { r } 
\frac { - 15 } { 4 } \
\frac { 5 } { 4 } \
0 \
\frac { 5 } { 2 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 35 } { 4 } \\
\frac { - 25 } { 4 } \\
\frac { 45 } { 4 } \\
\frac { - 45 } { 4 }
\end{array} \right]$$

Question 5

Use the inverse formula $$A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } 
d & - b \
- c & a
\end{array} \right]$$ to find the inverse of the 2×2 matrix (if it exists).​ $$\left[ \begin{array} { c c } 
6 & 7 \
- 5 & 13
\end{array} \right]$$ ​

Accepted Answer

A) ​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
13 & - 7 \
5 & 6
\end{array} \right]$$ 
B) ​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & 7 \
5 & - 6
\end{array} \right]$$ 
C) ​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & - 7 \
5 & 6
\end{array} \right]$$ 
D) ​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
13 & 7 \
5 & 6
\end{array} \right]$$ 
E) ​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & - 7 \
- 5 & - 6
\end{array} \right]$$ 
A) ​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
13 & - 7 \
5 & 6
\end{array} \right]$$ 
B) ​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & 7 \
5 & - 6
\end{array} \right]$$ 
C) ​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & - 7 \
5 & 6
\end{array} \right]$$ 
D) ​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
13 & 7 \
5 & 6
\end{array} \right]$$ 
E) ​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & - 7 \
- 5 & - 6
\end{array} \right]$$

Question 6

A small home business creates muffins,bones,and cookies for dogs.In addition to other ingredients,each muffin requires 2 units of beef,3 units of chicken,and 2 units of liver.Each bone requires 1 unit of beef,1 unit of chicken,and 1 unit of liver.Each cookie requires 2 units of beef,1 unit of chicken,and 1.5 units of liver.Find the numbers of muffins,bones,and cookies that the company can create with the given amounts of ingredients. ​
3,000 units of beef
2,950 units of chicken
2,900 units of liver
​

Accepted Answer

A) 200 muffins,2,300 bones,200 cookies 
B) 150 muffins,150 bones,2,950 cookies 
C) 200 muffins,2,300 bones,150 cookies 
D) 2,300 muffins,150 bones,200 cookies 
E) 150 muffins,2,300 bones,200 cookies 
A) 200 muffins,2,300 bones,200 cookies 
B) 150 muffins,150 bones,2,950 cookies 
C) 200 muffins,2,300 bones,150 cookies 
D) 2,300 muffins,150 bones,200 cookies 
E) 150 muffins,2,300 bones,200 cookies

Question 7

Find the inverse of the matrix $$\left[ \begin{array} { c c c } 
7 & 7 & 7 \
21 & 35 & 28 \
21 & 42 & 35
\end{array} \right]$$ . ​

Accepted Answer

A) ​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 1 & - 1 \
- 3 & 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ 
B) ​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 1 & 1 \
0 & - 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ 
C) ​ $$- 7 \left[ \begin{array} { c c c } 
3 & 1 & 0 \
- 3 & 1 & - 1 \
1 & - 3 & 2
\end{array} \right]$$ 
D) ​ $$- \frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 0 & - 1 \
- 3 & 2 & - 1 \
2 & - 3 & 3
\end{array} \right]$$ 
E) ​ $$7 \left[ \begin{array} { c c c } 
1 & 1 & 1 \
- 3 & 0 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ 
A) ​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 1 & - 1 \
- 3 & 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ 
B) ​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 1 & 1 \
0 & - 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$$ 
C) ​ $$- 7 \left[ \begin{array} { c c c } 
3 & 1 & 0 \
- 3 & 1 & - 1 \
1 & - 3 & 2
\end{array} \right]$$ 
D) ​ $$- \frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 0 & - 1 \
- 3 & 2 & - 1 \
2 & - 3 & 3
\end{array} \right]$$ 
E) ​ $$7 \left[ \begin{array} { c c c } 
1 & 1 & 1 \
- 3 & 0 & - 1 \
3 & - 3 & 2
\end{array} \right]$$

Question 8

Use the inverse formula $$A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } 
d & - b \
- c & a
\end{array} \right]$$ to find the inverse of the 2×2 matrix (if it exists). ​​ $$\left[ \begin{array} { l l } 
2 & 3 \
1 & 5
\end{array} \right]$$ ​

Accepted Answer

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

Question 9

Find the inverse of the matrix $$\left[ \begin{array} { c c } 
- 16 & 22 \
- 8 & 11
\end{array} \right]$$ (if it exists).

Accepted Answer

A) does not exist 
B) ​ $- \frac { 1 } { 16 }$ $\left[ \begin{array} { c c } 
11 & - 22 \
8 & - 16
\end{array} \right]$ 
C) ​ $\frac { 1 } { 11 }$ $\left[ \begin{array} { c c } 
- 16 & 22 \
- 8 & 11
\end{array} \right]$ 
D) ​ $- \frac { 1 } { 22 }$ $\left[ \begin{array} { c c } 
- 11 & 22 \
- 8 & 16
\end{array} \right]$ 
E) ​ $\frac { 1 } { 8 }$ $\left[ \begin{array} { c c } 
11 & - 22 \
8 & - 16
\end{array} \right]$ 
A) does not exist 
B) ​ $- \frac { 1 } { 16 }$ $\left[ \begin{array} { c c } 
11 & - 22 \
8 & - 16
\end{array} \right]$ 
C) ​ $\frac { 1 } { 11 }$ $\left[ \begin{array} { c c } 
- 16 & 22 \
- 8 & 11
\end{array} \right]$ 
D) ​ $- \frac { 1 } { 22 }$ $\left[ \begin{array} { c c } 
- 11 & 22 \
- 8 & 16
\end{array} \right]$ 
E) ​ $\frac { 1 } { 8 }$ $\left[ \begin{array} { c c } 
11 & - 22 \
8 & - 16
\end{array} \right]$

Question 10

Use the inverse formula $$A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } 
d & - b \
- c & a
\end{array} \right]$$ to find the inverse of the 2×2 matrix (if it exists). ​​ $$\left[ \begin{array} { c c } 
1 & - 2 \
- 3 & 2
\end{array} \right]$$ ​

Accepted Answer

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

Question 11

Solve the system of linear equations.​ $$\left\{ \begin{array} { c } 
x - 2 y = 6 \
2 x - 3 y = 4
\end{array} \right.$$ ​ ​

Accepted Answer

A) ​ $( - 10 , - 8 )$ 
B) ​ $( - 10,8 )$ 
C) ​ $( - 10,4 )$ 
D) ​ $( - 4 , - 10 )$ 
E) ​ $( 10,8 )$ 
A) ​ $( - 10 , - 8 )$ 
B) ​ $( - 10,8 )$ 
C) ​ $( - 10,4 )$ 
D) ​ $( - 4 , - 10 )$ 
E) ​ $( 10,8 )$

Question 12

​Solve the system of linear equations $$\left\{ \begin{array} { l } 
4 x + 4 y = 3 \
8 x - 20 y = - 9
\end{array} \right.$$ using an inverse matrix.

Accepted Answer

A) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 3 } { 14 } \\
\frac { 15 } { 28 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 9 } { 28 } \\
\frac { 12 } { 7 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 28 } \\
\frac { - 3 } { 14 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 7 } \\
\frac { 15 } { 28 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 9 } { 7 } \\
\frac { - 3 } { 14 }
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 3 } { 14 } \\
\frac { 15 } { 28 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 9 } { 28 } \\
\frac { 12 } { 7 }
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 28 } \\
\frac { - 3 } { 14 }
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 7 } \\
\frac { 15 } { 28 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 9 } { 7 } \\
\frac { - 3 } { 14 }
\end{array} \right]$$

Question 13

Find the inverse of the matrix $$\left[ \begin{array} { c c } 
- 2 & 1 \
5 & 4
\end{array} \right]$$ (if it exists).

Accepted Answer

A) ​ $$\frac { 1 } { 13 } \left[ \begin{array} { c c } 
4 & - 1 \
- 5 & - 2
\end{array} \right]$$ 
B) ​ $$- \frac { 1 } { 13 } \left[ \begin{array} { c c } 
4 & - 1 \
- 5 & - 2
\end{array} \right]$$ 
C) ​ $$- \frac { 1 } { 13 } \left[ \begin{array} { c c } 
- 4 & 1 \
5 & 2
\end{array} \right]$$ 
D) ​ $$- \frac { 1 } { 13 } \left[ \begin{array} { c c } 
- 2 & 1 \
5 & 4
\end{array} \right]$$ 
E) does not exist 
A) ​ $$\frac { 1 } { 13 } \left[ \begin{array} { c c } 
4 & - 1 \
- 5 & - 2
\end{array} \right]$$ 
B) ​ $$- \frac { 1 } { 13 } \left[ \begin{array} { c c } 
4 & - 1 \
- 5 & - 2
\end{array} \right]$$ 
C) ​ $$- \frac { 1 } { 13 } \left[ \begin{array} { c c } 
- 4 & 1 \
5 & 2
\end{array} \right]$$ 
D) ​ $$- \frac { 1 } { 13 } \left[ \begin{array} { c c } 
- 2 & 1 \
5 & 4
\end{array} \right]$$ 
E) does not exist

Question 14

Find the inverse of the matrix (if it exists).​ $$\left[ \begin{array} { l l } 
3 & 0 \
0 & 4
\end{array} \right]$$ ​

Accepted Answer

A) ​ $$\left[ \begin{array} { l l } 
\frac { 1 } { 4 } & 0 \
0 & \frac { 1 } { 4 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l l } 
\frac { 1 } { 4 } & 0 \
0 & 0
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
0 & \frac { 1 } { 4 } \
\frac { 1 } { 4 } & 0
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l l } 
\frac { 1 } { 4 } & 0 \
0 & \frac { 1 } { 3 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l l } 
\frac { 1 } { 3 } & 0 \
0 & \frac { 1 } { 4 }
\end{array} \right]$$ 
A) ​ $$\left[ \begin{array} { l l } 
\frac { 1 } { 4 } & 0 \
0 & \frac { 1 } { 4 }
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { l l } 
\frac { 1 } { 4 } & 0 \
0 & 0
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
0 & \frac { 1 } { 4 } \
\frac { 1 } { 4 } & 0
\end{array} \right]$$ 
D) ​ $$\left[ \begin{array} { l l } 
\frac { 1 } { 4 } & 0 \
0 & \frac { 1 } { 3 }
\end{array} \right]$$ 
E) ​ $$\left[ \begin{array} { l l } 
\frac { 1 } { 3 } & 0 \
0 & \frac { 1 } { 4 }
\end{array} \right]$$

Question 15

Solve the system of linear equations.​ $$\left\{ \begin{array} { l l } 
x - 2 y & = 5 \
2 x - 3 y & = 10
\end{array} \right.$$ ​

Accepted Answer

A) ​ $( 5,10 )$ 
B) ​ $( 5,5 )$ 
C) ​ $( 10,5 )$ 
D) ​ $( 5,0 )$ 
E) ​ $( 0,5 )$ 
A) ​ $( 5,10 )$ 
B) ​ $( 5,5 )$ 
C) ​ $( 10,5 )$ 
D) ​ $( 5,0 )$ 
E) ​ $( 0,5 )$

Question 16

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. Total Investment
Annual Return
$32,000
2465 $$\left\{ \begin{array} { c l } 
x + y + z & = 32,000 \text { (total investment) } \
0.065 x + 0.07 y + 0.09 z & = 2465 \text { (annual retur) } \
2 y - z & = 0
\end{array} \right.$$ Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond.
​

Accepted Answer

A) $11,000 in AAA-rated bonds $7,000 in A-rated bonds
$14,000 in B-rated bonds 
B) $11,000 in AAA-rated bonds $14,000 in A-rated bonds
$7,000 in B-rated bonds 
C) $14,000 in AAA-rated bonds $11,000 in A-rated bonds
$7,000 in B-rated bonds 
D) $14,000 in AAA-rated bonds $7,000 in A-rated bonds
$11,000 in B-rated bonds 
E) $7,000 in AAA-rated bonds $11,000 in A-rated bonds
$14,000 in B-rated bonds 
A) $11,000 in AAA-rated bonds $7,000 in A-rated bonds
$14,000 in B-rated bonds 
B) $11,000 in AAA-rated bonds $14,000 in A-rated bonds
$7,000 in B-rated bonds 
C) $14,000 in AAA-rated bonds $11,000 in A-rated bonds
$7,000 in B-rated bonds 
D) $14,000 in AAA-rated bonds $7,000 in A-rated bonds
$11,000 in B-rated bonds 
E) $7,000 in AAA-rated bonds $11,000 in A-rated bonds
$14,000 in B-rated bonds

Question 17

Use the inverse formula $$A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } 
d & - b \
- c & a
\end{array} \right]$$ to find the inverse of the matrix (if it exists). ​​ $$\left[ \begin{array} { c c } 
- 7 & - 9 \
7 & 9
\end{array} \right]$$ ​

Accepted Answer

A) ​ $$\left[ \begin{array} { c c } 
- 9 & - 9 \
7 & 7
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { r r } 
- 9 & - 9 \
- 7 & - 7
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
- 9 & 9 \
7 & - 7
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c } 
9 & - 9 \
7 & 7
\end{array} \right]$$ 
E) ​Does not exist 
A) ​ $$\left[ \begin{array} { c c } 
- 9 & - 9 \
7 & 7
\end{array} \right]$$ 
B) ​ $$\left[ \begin{array} { r r } 
- 9 & - 9 \
- 7 & - 7
\end{array} \right]$$ 
C) ​ $$\left[ \begin{array} { c c } 
- 9 & 9 \
7 & - 7
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c } 
9 & - 9 \
7 & 7
\end{array} \right]$$ 
E) ​Does not exist

Question 18

Solve the system of linear equations.​ $$\left\{ \begin{array} { c } 
x + y + z = - 7 \
3 x + 5 y + 4 z = 8 \
3 x + 6 y + 5 z = 0
\end{array} \right.$$ ​

Accepted Answer

A) ​ $( 2,37 , - 45 )$ 
B) ​ $( - 1,37 , - 45 )$ 
C) ​ $( - 1 , - 37 , - 45 )$ 
D) ​ $( 1,37 , - 45 )$ 
E) ​ $( 1,38 , - 45 )$ 
A) ​ $( 2,37 , - 45 )$ 
B) ​ $( - 1,37 , - 45 )$ 
C) ​ $( - 1 , - 37 , - 45 )$ 
D) ​ $( 1,37 , - 45 )$ 
E) ​ $( 1,38 , - 45 )$

Question 19

Show that B is the inverse of A.Show all your work. $$A = \left[ \begin{array} { c c } 
2 & 9 \
- 4 & 3
\end{array} \right] , B = \left[ \begin{array} { c c } 
\frac { 1 } { 14 } & - \frac { 3 } { 14 } \
\frac { 2 } { 21 } & \frac { 1 } { 21 }
\end{array} \right]$$ ​

Accepted Answer

The answer of Show that B is the inverse of...

Question 20

The table shows the enrollment projections (in millions)for public universities in the United States for the years 2010 through 2012.
Year
Enrollment projections
2010
13)83
2011
14)07
2012
14)23
The data can be modeled by the quadratic function $y=a t^{2}+b t+c$ .Create a system of linear equations for the data.Let t represent the year,with t = 10 corresponding to 2010.

Accepted Answer

A)  $$\left\{ \begin{array} { l } 
100 a + 10 b + c = 13.83 \
121 a - 11 b + c = 14.07 \
144 a + 12 b + c = 14.23
\end{array} \right.$$ 
B) ​ $$\left\{ \begin{array} { l } 
100 a - 10 b + c = 13.83 \
121 a + 11 b + c = 14.07 \
144 a + 12 b + c = 14.23
\end{array} \right.$$ 
C) ​ $$\left\{ \begin{array} { l } 
100 a + 10 b + c = 13.83 \
121 a + 11 b + c = 14.07 \
144 a + 12 b + c = 14.23
\end{array} \right.$$ 
D) ​ $$\left\{ \begin{array} { l } 
100 a + 10 b + c = 13.83 \
121 a + 11 b + c = 14.07 \
144 a + 12 b - c = 14.23
\end{array} \right.$$ 
E) ​ $$\left\{ \begin{array} { l } 
100 a - 10 b - c = 13.83 \
121 a - 11 b - c = 14.07 \
144 a - 12 b - c = 14.23
\end{array} \right.$$ 
A)  $$\left\{ \begin{array} { l } 
100 a + 10 b + c = 13.83 \
121 a - 11 b + c = 14.07 \
144 a + 12 b + c = 14.23
\end{array} \right.$$ 
B) ​ $$\left\{ \begin{array} { l } 
100 a - 10 b + c = 13.83 \
121 a + 11 b + c = 14.07 \
144 a + 12 b + c = 14.23
\end{array} \right.$$ 
C) ​ $$\left\{ \begin{array} { l } 
100 a + 10 b + c = 13.83 \
121 a + 11 b + c = 14.07 \
144 a + 12 b + c = 14.23
\end{array} \right.$$ 
D) ​ $$\left\{ \begin{array} { l } 
100 a + 10 b + c = 13.83 \
121 a + 11 b + c = 14.07 \
144 a + 12 b - c = 14.23
\end{array} \right.$$ 
E) ​ $$\left\{ \begin{array} { l } 
100 a - 10 b - c = 13.83 \
121 a - 11 b - c = 14.07 \
144 a - 12 b - c = 14.23
\end{array} \right.$$

Exam 49: The Inverse of a Square Matrix

Solve the system of linear equations.​ {x−2y=32x−3y=−4\left\{ \begin{array} { l } x - 2 y = 3 \\2 x - 3 y = - 4\end{array} \right.{x−2y=32x−3y=−4​ ​

​Use an inverse matrix to solve (if possible)the system of linear equations.​ {1.8x−5y=428.8x−16y=5\left\{ \begin{array} { l l } 1.8 x - 5 y & = 4 \\28.8 x - 16 y & = 5\end{array} \right.{1.8x−5y28.8x−16y​=4=5​ ​

Find the inverse of A.​ A=[−41248200124−8]A = \left[ \begin{array} { c c c } - 4 & 12 & 4 \\8 & 20 & 0 \\12 & 4 & - 8\end{array} \right]A=​−4812​12204​40−8​​ ​

Find the inverse of the matrix [777213528214235]\left[ \begin{array} { c c c } 7 & 7 & 7 \\21 & 35 & 28 \\21 & 42 & 35\end{array} \right]​72121​73542​72835​​ . ​

Find the inverse of the matrix [−1622−811]\left[ \begin{array} { c c } - 16 & 22 \\- 8 & 11\end{array} \right][−16−8​2211​] (if it exists).

Solve the system of linear equations.​ {x−2y=62x−3y=4\left\{ \begin{array} { c } x - 2 y = 6 \\2 x - 3 y = 4\end{array} \right.{x−2y=62x−3y=4​ ​ ​

​Solve the system of linear equations {4x+4y=38x−20y=−9\left\{ \begin{array} { l } 4 x + 4 y = 3 \\8 x - 20 y = - 9\end{array} \right.{4x+4y=38x−20y=−9​ using an inverse matrix.

Find the inverse of the matrix [−2154]\left[ \begin{array} { c c } - 2 & 1 \\5 & 4\end{array} \right][−25​14​] (if it exists).

Find the inverse of the matrix (if it exists).​ [3004]\left[ \begin{array} { l l } 3 & 0 \\0 & 4\end{array} \right][30​04​] ​

Solve the system of linear equations.​ {x−2y=52x−3y=10\left\{ \begin{array} { l l } x - 2 y & = 5 \\2 x - 3 y & = 10\end{array} \right.{x−2y2x−3y​=5=10​ ​

Solve the system of linear equations.​ {x+y+z=−73x+5y+4z=83x+6y+5z=0\left\{ \begin{array} { c } x + y + z = - 7 \\3 x + 5 y + 4 z = 8 \\3 x + 6 y + 5 z = 0\end{array} \right.⎩⎨⎧​x+y+z=−73x+5y+4z=83x+6y+5z=0​ ​

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

Polar Equations of Conics

Graphs of Polar Equations

Filters

Solve the system of linear equations. $\left\{ \begin{array} { l } x - 2 y = 3 \\2 x - 3 y = - 4\end{array} \right.$

Use an inverse matrix to solve (if possible)the system of linear equations. $\left\{ \begin{array} { l l } 1.8 x - 5 y & = 4 \\28.8 x - 16 y & = 5\end{array} \right.$

Find the inverse of A. $A = \left[ \begin{array} { c c c } - 4 & 12 & 4 \\8 & 20 & 0 \\12 & 4 & - 8\end{array} \right]$

Find the inverse of the matrix $\left[ \begin{array} { c c c } 7 & 7 & 7 \\21 & 35 & 28 \\21 & 42 & 35\end{array} \right]$ .

Find the inverse of the matrix $\left[ \begin{array} { c c } - 16 & 22 \\- 8 & 11\end{array} \right]$ (if it exists).

Solve the system of linear equations. $\left\{ \begin{array} { c } x - 2 y = 6 \\2 x - 3 y = 4\end{array} \right.$

Solve the system of linear equations $\left\{ \begin{array} { l } 4 x + 4 y = 3 \\8 x - 20 y = - 9\end{array} \right.$ using an inverse matrix.

Find the inverse of the matrix $\left[ \begin{array} { c c } - 2 & 1 \\5 & 4\end{array} \right]$ (if it exists).

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { l l } 3 & 0 \\0 & 4\end{array} \right]$

Solve the system of linear equations. $\left\{ \begin{array} { l l } x - 2 y & = 5 \\2 x - 3 y & = 10\end{array} \right.$

Solve the system of linear equations. $\left\{ \begin{array} { c } x + y + z = - 7 \\3 x + 5 y + 4 z = 8 \\3 x + 6 y + 5 z = 0\end{array} \right.$