Question 1

Find A + B.​ $$A = \left[ \begin{array} { l l } 
1 & 2 \
2 & 1
\end{array} \right] , B = \left[ \begin{array} { c c } 
- 4 & - 4 \
5 & 4
\end{array} \right]$$ ​

Accepted Answer

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

Question 2

Use Gauss-Jordan elimination on the augmented matrix [A|B] to solve for the matrix X.​ $$\left\{ \begin{aligned}
- x _ { 1 } + x _ { 2 } & = 7 \
- 2 x _ { 1 } + x _ { 2 } & = 0
\end{aligned} \right.$$ ​

Accepted Answer

A)  $$\left[ \begin{array} { c } 
- 7 \
- 14
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c } 
7 \
14
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c } 
7 \
- 14
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c } 
- 7 \
14
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c } 
14 \
7
\end{array} \right]$$ 
A)  $$\left[ \begin{array} { c } 
- 7 \
- 14
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c } 
7 \
14
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c } 
7 \
- 14
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c } 
- 7 \
14
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c } 
14 \
7
\end{array} \right]$$ B

Question 3

Solve the X in the following equation. ​​ $$A = \left[ \begin{array} { c c } 
- 3 & - 3 \
2 & 5 \
- 3 & 6
\end{array} \right] \text { and } B = \left[ \begin{array} { c c } 
2 & 4 \
- 1 & - 2 \
1 & 12
\end{array} \right]$$ ​
X = 3A - 2B
​

Accepted Answer

A)  $$\left[ \begin{array} { c c } 
- 13 & - 17 \
8 & 19 \
- 11 & - 6
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c } 
- 11 & - 17 \
8 & - 6 \
- 11 & 19
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c } 
- 9 & - 17 \
- 11 & 19 \
8 & - 6
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c } 
- 15 & 19 \
8 & - 17 \
- 11 & - 6
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c } 
- 17 & - 17 \
19 & 8 \
- 11 & - 6
\end{array} \right]$$ 
A)  $$\left[ \begin{array} { c c } 
- 13 & - 17 \
8 & 19 \
- 11 & - 6
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c } 
- 11 & - 17 \
8 & - 6 \
- 11 & 19
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c } 
- 9 & - 17 \
- 11 & 19 \
8 & - 6
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c } 
- 15 & 19 \
8 & - 17 \
- 11 & - 6
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c } 
- 17 & - 17 \
19 & 8 \
- 11 & - 6
\end{array} \right]$$ A

Question 4

Find A - B.​ $$A = \left[ \begin{array} { l l l } 
4 & 9 & 2 \
6 & 3 & 2 \
9 & 6 & 8
\end{array} \right] , B = \left[ \begin{array} { l l l } 
3 & 9 & 9 \
6 & 8 & 7 \
7 & 7 & 8
\end{array} \right]$$ ​

Accepted Answer

A)  $$\left[ \begin{array} { c c c } 
1 & 0 & - 7 \
0 & - 5 & - 5 \
9 & 6 & 8
\end{array} \right]$$ ​ 
B)  $$\left[ \begin{array} { c c c } 
4 & 9 & 2 \
6 & 8 & 7 \
2 & - 1 & 0
\end{array} \right]$$ ​ 
C)  $$\left[ \begin{array} { c c c } 
1 & 0 & - 7 \
6 & 8 & 7 \
2 & - 1 & 0
\end{array} \right]$$ ​ 
D)  $$\left[ \begin{array} { c c c } 
1 & 0 & - 7 \
0 & - 5 & - 5 \
2 & - 1 & 8
\end{array} \right]$$ ​ 
E)  $$\left[ \begin{array} { c c c } 
1 & 0 & - 7 \
0 & - 5 & - 5 \
2 & - 1 & 0
\end{array} \right]$$ ​ 
A)  $$\left[ \begin{array} { c c c } 
1 & 0 & - 7 \
0 & - 5 & - 5 \
9 & 6 & 8
\end{array} \right]$$ ​ 
B)  $$\left[ \begin{array} { c c c } 
4 & 9 & 2 \
6 & 8 & 7 \
2 & - 1 & 0
\end{array} \right]$$ ​ 
C)  $$\left[ \begin{array} { c c c } 
1 & 0 & - 7 \
6 & 8 & 7 \
2 & - 1 & 0
\end{array} \right]$$ ​ 
D)  $$\left[ \begin{array} { c c c } 
1 & 0 & - 7 \
0 & - 5 & - 5 \
2 & - 1 & 8
\end{array} \right]$$ ​ 
E)  $$\left[ \begin{array} { c c c } 
1 & 0 & - 7 \
0 & - 5 & - 5 \
2 & - 1 & 0
\end{array} \right]$$ ​

Question 5

Write the system of linear equations as a matrix equation,AX = B.​ $$\left\{ \begin{array} { l } 
2 x _ { 1 } + 3 x _ { 2 } = 8 \
4 x _ { 1 } + 5 x _ { 2 } = 16
\end{array} \right.$$ ​

Accepted Answer

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

Question 6

​Evaluate the expression. ​​ $$\frac { 3 } { 2 } \left[ \begin{array} { l l l } 
- 8 & - 1 & - 1
\end{array} \right] + \left[ \begin{array} { l l l } 
- 9 & - 3 & 3
\end{array} \right]$$

Accepted Answer

A) ​[-21 $- \frac { 9 } { 2 }$ $\frac { 3 } { 2 }$ ] 
B) ​[-17 -4 2] 
C) ​[-3 $\frac { 3 } { 2 }$ $- \frac { 9 } { 2 }$ ] 
D) ​[ $- \frac { 33 } { 2 }$ -3 0] 
E) ​not possible 
A) ​[-21 $- \frac { 9 } { 2 }$ $\frac { 3 } { 2 }$ ] 
B) ​[-17 -4 2] 
C) ​[-3 $\frac { 3 } { 2 }$ $- \frac { 9 } { 2 }$ ] 
D) ​[ $- \frac { 33 } { 2 }$ -3 0] 
E) ​not possible

Question 7

Find A - B.​ $$A = \left[ \begin{array} { c c } 
4 & - 4 \
4 & 5 \
- 3 & 4
\end{array} \right] , B = \left[ \begin{array} { c c } 
3 & 8 \
- 3 & - 5 \
3 & 7
\end{array} \right]$$ ​

Accepted Answer

A)  $$\left[ \begin{array} { c c } 
1 & - 12 \
7 & 10 \
- 6 & - 3
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c } 
3 & - 12 \
7 & - 3 \
- 6 & 10
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c } 
5 & - 12 \
- 6 & 10 \
7 & - 3
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c } 
- 1 & 10 \
7 & - 12 \
- 6 & - 3
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c } 
- 3 & - 12 \
10 & 7 \
- 6 & - 3
\end{array} \right]$$ 
A)  $$\left[ \begin{array} { c c } 
1 & - 12 \
7 & 10 \
- 6 & - 3
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c } 
3 & - 12 \
7 & - 3 \
- 6 & 10
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c } 
5 & - 12 \
- 6 & 10 \
7 & - 3
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c } 
- 1 & 10 \
7 & - 12 \
- 6 & - 3
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c } 
- 3 & - 12 \
10 & 7 \
- 6 & - 3
\end{array} \right]$$

Question 8

Find the product.​ $$\left[ \begin{array} { c c c c } 
2 & 1 & 0 & 0 \
- 1 & 2 & 0 & - 4 \
0 & 0 & 2 & 0 \
0 & 4 & 0 & 2
\end{array} \right] \left[ \begin{array} { c } 
5 \
3 \
- 3 \
- 5
\end{array} \right]$$ ​

Accepted Answer

A)  $$\left[ \begin{array} { c } 
13 \
21 \
- 6 \
2
\end{array} \right]$$ ​ 
B)  $$\left[ \begin{array} { c c } 
13 & 21 \
- 6 & 2
\end{array} \right]$$ ​ 
C)  $$\left[ \begin{array} { c } 
2 \
- 6 \
21 \
13
\end{array} \right]$$ ​ 
D)  $$\left[ \begin{array} { l l l l } 
13 & 21 & - 6 & 2
\end{array} \right]$$ ​ 
E) cannot multiply ​ 
A)  $$\left[ \begin{array} { c } 
13 \
21 \
- 6 \
2
\end{array} \right]$$ ​ 
B)  $$\left[ \begin{array} { c c } 
13 & 21 \
- 6 & 2
\end{array} \right]$$ ​ 
C)  $$\left[ \begin{array} { c } 
2 \
- 6 \
21 \
13
\end{array} \right]$$ ​ 
D)  $$\left[ \begin{array} { l l l l } 
13 & 21 & - 6 & 2
\end{array} \right]$$ ​ 
E) cannot multiply ​

Question 9

If possible,find A2. (Note: A2 = AA. )​ $$A = \left[ \begin{array} { l l } 
4 & 5 \
4 & 4
\end{array} \right]$$ ​

Accepted Answer

A)  $$\left[ \begin{array} { l l } 
34 & 40 \
32 & 36
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { l l } 
36 & 40 \
32 & 36
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { l l } 
32 & 40 \
32 & 36
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { l l } 
38 & 40 \
32 & 36
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { l l } 
40 & 40 \
32 & 36
\end{array} \right]$$ 
A)  $$\left[ \begin{array} { l l } 
34 & 40 \
32 & 36
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { l l } 
36 & 40 \
32 & 36
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { l l } 
32 & 40 \
32 & 36
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { l l } 
38 & 40 \
32 & 36
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { l l } 
40 & 40 \
32 & 36
\end{array} \right]$$

Question 10

Find x and y. $$\left[ \begin{array} { c c c } 
x - 7 & 3 & - 8 y \
4 & - 8 x & - 7 \
- 9 & y - 3 & - 3
\end{array} \right] = \left[ \begin{array} { c c c } 
7 x - 19 & 3 & 24 \
4 & - 16 & - 7 \
- 9 & - 6 & - 3
\end{array} \right]$$

Accepted Answer

A) x = -3,y = 3 
B) x = 3,y = -4 
C) no solution 
D) x = 2,y = -3 
E) x = 2,y = -4 
A) x = -3,y = 3 
B) x = 3,y = -4 
C) no solution 
D) x = 2,y = -3 
E) x = 2,y = -4

Question 11

Find 3A.​ $$A = \left[ \begin{array} { c c c } 
- 2 & 5 & 0 \
5 & - 2 & 4 \
6 & 8 & - 1 \
0 & 6 & - 6 \
- 2 & - 1 & 0
\end{array} \right]$$ ​

Accepted Answer

A)  $$\left[ \begin{array} { c c c } 
- 8 & 15 & 0 \
0 & - 6 & - 18 \
18 & 24 & - 3 \
15 & 18 & 12 \
- 6 & - 3 & 0
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c c } 
- 6 & 15 & 0 \
15 & - 6 & 12 \
18 & 24 & - 3 \
0 & 18 & - 18 \
- 6 & - 3 & 0
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c c } 
- 10 & 15 & 0 \
15 & - 6 & 12 \
- 6 & 24 & 0 \
0 & 18 & - 18 \
18 & - 3 & - 3
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c c } 
- 4 & 15 & 0 \
15 & - 6 & 12 \
- 3 & 24 & 18 \
0 & 18 & - 18 \
0 & - 3 & - 6
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c c } 
- 2 & 15 & 0 \
12 & - 6 & 15 \
18 & 24 & - 3 \
- 18 & 18 & 0 \
- 6 & - 3 & 0
\end{array} \right]$$ 
A)  $$\left[ \begin{array} { c c c } 
- 8 & 15 & 0 \
0 & - 6 & - 18 \
18 & 24 & - 3 \
15 & 18 & 12 \
- 6 & - 3 & 0
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c c } 
- 6 & 15 & 0 \
15 & - 6 & 12 \
18 & 24 & - 3 \
0 & 18 & - 18 \
- 6 & - 3 & 0
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c c } 
- 10 & 15 & 0 \
15 & - 6 & 12 \
- 6 & 24 & 0 \
0 & 18 & - 18 \
18 & - 3 & - 3
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c c } 
- 4 & 15 & 0 \
15 & - 6 & 12 \
- 3 & 24 & 18 \
0 & 18 & - 18 \
0 & - 3 & - 6
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c c } 
- 2 & 15 & 0 \
12 & - 6 & 15 \
18 & 24 & - 3 \
- 18 & 18 & 0 \
- 6 & - 3 & 0
\end{array} \right]$$

Question 12

Write the system of linear equations as a matrix equation AX = B,and use Gauss-Jordan elimination on the augmented matrix $[ A \mid B ]$ to solve for the matrix X. $\left\{ \begin{aligned}
x _ { 1 } - 3 x _ { 2 } & = - 25 \
- 5 x _ { 1 } + 2 x _ { 2 } & = 34
\end{aligned} \right.$

Accepted Answer

A)  $$\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 25 \
- 4
\end{array} \right] = \left[ \begin{array} { c } 
34 \
7
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] = \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 4 \
- 25
\end{array} \right] = \left[ \begin{array} { c } 
7 \
34
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 4 \
7
\end{array} \right] = \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right] = \left[ \begin{array} { c } 
- 4 \
7
\end{array} \right]$$ 
A)  $$\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 25 \
- 4
\end{array} \right] = \left[ \begin{array} { c } 
34 \
7
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] = \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 4 \
- 25
\end{array} \right] = \left[ \begin{array} { c } 
7 \
34
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 4 \
7
\end{array} \right] = \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right] = \left[ \begin{array} { c } 
- 4 \
7
\end{array} \right]$$

Question 13

Solve for X in the equation given. $$5 A + 15 B = 5 X , A = \left[ \begin{array} { c c c } 
- 9 & 7 & - 4 \
- 5 & - 2 & - 2
\end{array} \right] \text { and } B = \left[ \begin{array} { c c c } 
- 2 & - 1 & - 4 \
- 5 & - 2 & 7
\end{array} \right]$$

Accepted Answer

A)  $$\left[ \begin{array} { c c c } 
- 15 & 4 & - 16 \
- 20 & - 8 & 19
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c c } 
- 11 & 6 & - 8 \
- 10 & - 4 & 5
\end{array} \right]$$ 
C) not possible 
D)  $$\left[ \begin{array} { c c c } 
- 75 & 20 & - 80 \
- 100 & - 40 & 95
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c c } 
- 3 & 10 & 8 \
10 & 4 & - 23
\end{array} \right]$$ 
A)  $$\left[ \begin{array} { c c c } 
- 15 & 4 & - 16 \
- 20 & - 8 & 19
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c c } 
- 11 & 6 & - 8 \
- 10 & - 4 & 5
\end{array} \right]$$ 
C) not possible 
D)  $$\left[ \begin{array} { c c c } 
- 75 & 20 & - 80 \
- 100 & - 40 & 95
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c c } 
- 3 & 10 & 8 \
10 & 4 & - 23
\end{array} \right]$$

Question 14

Find A - B.​ $$A = \left[ \begin{array} { l l } 
2 & - 2 \
4 & - 2
\end{array} \right] , B = \left[ \begin{array} { c c } 
4 & - 2 \
- 2 & 8
\end{array} \right]$$ ​

Accepted Answer

A)  $\left[ \begin{array} { c c } 
2 & 0 \
- 10 & 6
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
0 & - 10 \
6 & 0
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
- 6 & 0 \
6 & - 10
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
- 4 & 0 \
6 & - 10
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c } 
- 2 & 0 \
6 & - 10
\end{array} \right]$ 
A)  $\left[ \begin{array} { c c } 
2 & 0 \
- 10 & 6
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
0 & - 10 \
6 & 0
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
- 6 & 0 \
6 & - 10
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
- 4 & 0 \
6 & - 10
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c } 
- 2 & 0 \
6 & - 10
\end{array} \right]$

Question 15

Solve for X in the equation given. $$3 X = - 5 A - B , A = \left[ \begin{array} { c c } 
9 & 1 \
- 3 & - 8
\end{array} \right] \text { and } B = \left[ \begin{array} { c c } 
- 33 & 4 \
30 & 46
\end{array} \right]$$

Accepted Answer

A)  $$\left[ \begin{array} { l l } 
- 4 & - 3 \
- 5 & - 2
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c } 
- 4 & 3 \
5 & - 2
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { l l } 
- 12 & - 9 \
- 15 & - 6
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { l l } 
1 & 0 \
0 & 1
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { l l } 
- 4 & 0 \
- 5 & 4
\end{array} \right]$$ 
A)  $$\left[ \begin{array} { l l } 
- 4 & - 3 \
- 5 & - 2
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c } 
- 4 & 3 \
5 & - 2
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { l l } 
- 12 & - 9 \
- 15 & - 6
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { l l } 
1 & 0 \
0 & 1
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { l l } 
- 4 & 0 \
- 5 & 4
\end{array} \right]$$

Question 16

A corporation has four factories,each of which manufactures sport utility vehicles and pickup trucks.The number of units of vehicle i produced at factory j in one day is represented by aij in the matrix​ $$A = \left[ \begin{array} { c c c c } 
90 & 100 & 80 & 50 \
100 & 100 & 70 & 50
\end{array} \right]$$ . ​
Find the production levels if production is increased by 10%.
​

Accepted Answer

A)  $$\left[ \begin{array} { c c c c } 
99 & 110 & 88 & 55 \
110 & 110 & 77 & 55
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { l l l l } 
113 & 110 & 88 & 55 \
110 & 110 & 77 & 55
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { l l l l } 
118 & 110 & 88 & 55 \
110 & 110 & 77 & 55
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { l l l l } 
123 & 110 & 88 & 55 \
110 & 110 & 77 & 55
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { l l l l } 
128 & 110 & 88 & 55 \
110 & 110 & 77 & 55
\end{array} \right]$$ 
A)  $$\left[ \begin{array} { c c c c } 
99 & 110 & 88 & 55 \
110 & 110 & 77 & 55
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { l l l l } 
113 & 110 & 88 & 55 \
110 & 110 & 77 & 55
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { l l l l } 
118 & 110 & 88 & 55 \
110 & 110 & 77 & 55
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { l l l l } 
123 & 110 & 88 & 55 \
110 & 110 & 77 & 55
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { l l l l } 
128 & 110 & 88 & 55 \
110 & 110 & 77 & 55
\end{array} \right]$$

Question 17

Find x and y.​ $$\left[ \begin{array} { c c } 
x & - 2 \
7 & y
\end{array} \right] = \left[ \begin{array} { c c } 
- 20 & - 2 \
7 & 3
\end{array} \right]$$ ​

Accepted Answer

A)  $x = 20 , y = - 3$ 
B)  $x = \frac { 1 } { 20 } , y = - \frac { 1 } { 3 }$ 
C)  $x = \frac { 1 } { 20 } , y = \frac { 1 } { 3 }$ 
D)  $x = - 20 , y = 3$ 
E)  $x = - \frac { 1 } { 20 } , y = \frac { 1 } { 3 }$ 
A)  $x = 20 , y = - 3$ 
B)  $x = \frac { 1 } { 20 } , y = - \frac { 1 } { 3 }$ 
C)  $x = \frac { 1 } { 20 } , y = \frac { 1 } { 3 }$ 
D)  $x = - 20 , y = 3$ 
E)  $x = - \frac { 1 } { 20 } , y = \frac { 1 } { 3 }$

Question 18

Find 6A.​ $$A = \left[ \begin{array} { l l } 
1 & - 1 \
1 & - 1
\end{array} \right]$$ ​

Accepted Answer

A)  $\left[ \begin{array} { l l } 
6 & - 6 \
6 & - 6
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 
8 & - 6 \
6 & - 6
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
10 & - 6 \
- 6 & 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { l l } 
4 & - 6 \
6 & - 6
\end{array} \right]$ 
E)  $\left[ \begin{array} { l l } 
2 & - 6 \
6 & - 6
\end{array} \right]$ 
A)  $\left[ \begin{array} { l l } 
6 & - 6 \
6 & - 6
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 
8 & - 6 \
6 & - 6
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
10 & - 6 \
- 6 & 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { l l } 
4 & - 6 \
6 & - 6
\end{array} \right]$ 
E)  $\left[ \begin{array} { l l } 
2 & - 6 \
6 & - 6
\end{array} \right]$

Question 19

Find the product.​ $$\left[ \begin{array} { c } 
3 \
- 6 \
- 9
\end{array} \right] \left[ \begin{array} { l l l } 
5 & - 9 & - 5
\end{array} \right]$$ ​

Accepted Answer

A)  $$\left[ \begin{array} { c c c } 
3 & 9 & - 15 \
6 & 5 & 30 \
9 & 9 & 45
\end{array} \right]$$ ​ 
B)  $$\left[ \begin{array} { c c c } 
15 & - 27 & - 15 \
- 30 & 54 & - 30 \
- 15 & - 27 & 15
\end{array} \right]$$ ​ 
C)  $$\left[ \begin{array} { c c c } 
15 & - 27 & - 15 \
- 30 & 54 & 30 \
- 45 & 81 & 45
\end{array} \right]$$ ​ 
D)  $$\left[ \begin{array} { l l l } 
15 & 54 & 45
\end{array} \right]$$ ​ 
E) 114 ​ 
A)  $$\left[ \begin{array} { c c c } 
3 & 9 & - 15 \
6 & 5 & 30 \
9 & 9 & 45
\end{array} \right]$$ ​ 
B)  $$\left[ \begin{array} { c c c } 
15 & - 27 & - 15 \
- 30 & 54 & - 30 \
- 15 & - 27 & 15
\end{array} \right]$$ ​ 
C)  $$\left[ \begin{array} { c c c } 
15 & - 27 & - 15 \
- 30 & 54 & 30 \
- 45 & 81 & 45
\end{array} \right]$$ ​ 
D)  $$\left[ \begin{array} { l l l } 
15 & 54 & 45
\end{array} \right]$$ ​ 
E) 114 ​

Question 20

Evaluate the expression.​ $$\left[ \begin{array} { c c } 
- 4 & 0 \
4 & - 4
\end{array} \right] + \left[ \begin{array} { c c } 
9 & 1 \
- 4 & - 1
\end{array} \right] + \left[ \begin{array} { c c } 
- 11 & - 12 \
13 & 7
\end{array} \right]$$ ​

Accepted Answer

A)  $$\left[ \begin{array} { c c } 
- 6 & - 11 \
13 & 2
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c } 
- 4 & - 11 \
2 & 13
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c } 
- 2 & 2 \
13 & - 11
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c } 
- 8 & 13 \
- 11 & 2
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c } 
- 10 & - 11 \
13 & 2
\end{array} \right]$$ 
A)  $$\left[ \begin{array} { c c } 
- 6 & - 11 \
13 & 2
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c } 
- 4 & - 11 \
2 & 13
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c } 
- 2 & 2 \
13 & - 11
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c } 
- 8 & 13 \
- 11 & 2
\end{array} \right]$$ 
E)  $$\left[ \begin{array} { c c } 
- 10 & - 11 \
13 & 2
\end{array} \right]$$

Exam 48: Operations With Matrices

Find A + B.​ A=[1221],B=[−4−454]A = \left[ \begin{array} { l l } 1 & 2 \\2 & 1\end{array} \right] , B = \left[ \begin{array} { c c } - 4 & - 4 \\5 & 4\end{array} \right]A=[12​21​],B=[−45​−44​] ​

Use Gauss-Jordan elimination on the augmented matrix [A|B] to solve for the matrix X.​ {−x1+x2=7−2x1+x2=0\left\{ \begin{aligned}- x _ { 1 } + x _ { 2 } & = 7 \\- 2 x _ { 1 } + x _ { 2 } & = 0\end{aligned} \right.{−x1​+x2​−2x1​+x2​​=7=0​ ​

Find A - B.​ A=[492632968],B=[399687778]A = \left[ \begin{array} { l l l } 4 & 9 & 2 \\6 & 3 & 2 \\9 & 6 & 8\end{array} \right] , B = \left[ \begin{array} { l l l } 3 & 9 & 9 \\6 & 8 & 7 \\7 & 7 & 8\end{array} \right]A=​469​936​228​​,B=​367​987​978​​ ​

Write the system of linear equations as a matrix equation,AX = B.​ {2x1+3x2=84x1+5x2=16\left\{ \begin{array} { l } 2 x _ { 1 } + 3 x _ { 2 } = 8 \\4 x _ { 1 } + 5 x _ { 2 } = 16\end{array} \right.{2x1​+3x2​=84x1​+5x2​=16​ ​

​Evaluate the expression. ​​ 32[−8−1−1]+[−9−33]\frac { 3 } { 2 } \left[ \begin{array} { l l l } - 8 & - 1 & - 1\end{array} \right] + \left[ \begin{array} { l l l } - 9 & - 3 & 3\end{array} \right]23​[−8​−1​−1​]+[−9​−3​3​]

Find A - B.​ A=[4−445−34],B=[38−3−537]A = \left[ \begin{array} { c c } 4 & - 4 \\4 & 5 \\- 3 & 4\end{array} \right] , B = \left[ \begin{array} { c c } 3 & 8 \\- 3 & - 5 \\3 & 7\end{array} \right]A=​44−3​−454​​,B=​3−33​8−57​​ ​

If possible,find A2. (Note: A2 = AA. )​ A=[4544]A = \left[ \begin{array} { l l } 4 & 5 \\4 & 4\end{array} \right]A=[44​54​] ​

Find 3A.​ A=[−2505−2468−106−6−2−10]A = \left[ \begin{array} { c c c } - 2 & 5 & 0 \\5 & - 2 & 4 \\6 & 8 & - 1 \\0 & 6 & - 6 \\- 2 & - 1 & 0\end{array} \right]A=​−2560−2​5−286−1​04−1−60​​ ​

Find A - B.​ A=[2−24−2],B=[4−2−28]A = \left[ \begin{array} { l l } 2 & - 2 \\4 & - 2\end{array} \right] , B = \left[ \begin{array} { c c } 4 & - 2 \\- 2 & 8\end{array} \right]A=[24​−2−2​],B=[4−2​−28​] ​

Find x and y.​ [x−27y]=[−20−273]\left[ \begin{array} { c c } x & - 2 \\7 & y\end{array} \right] = \left[ \begin{array} { c c } - 20 & - 2 \\7 & 3\end{array} \right][x7​−2y​]=[−207​−23​] ​

Find 6A.​ A=[1−11−1]A = \left[ \begin{array} { l l } 1 & - 1 \\1 & - 1\end{array} \right]A=[11​−1−1​] ​

Find the product.​ [3−6−9][5−9−5]\left[ \begin{array} { c } 3 \\- 6 \\- 9\end{array} \right] \left[ \begin{array} { l l l } 5 & - 9 & - 5\end{array} \right]​3−6−9​​[5​−9​−5​] ​

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

The Inverse of a Square Matrix

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

Find A + B. $A = \left[ \begin{array} { l l } 1 & 2 \\2 & 1\end{array} \right] , B = \left[ \begin{array} { c c } - 4 & - 4 \\5 & 4\end{array} \right]$

Use Gauss-Jordan elimination on the augmented matrix [A|B] to solve for the matrix X. $\left\{ \begin{aligned}- x _ { 1 } + x _ { 2 } & = 7 \\- 2 x _ { 1 } + x _ { 2 } & = 0\end{aligned} \right.$

Find A - B. $A = \left[ \begin{array} { l l l } 4 & 9 & 2 \\6 & 3 & 2 \\9 & 6 & 8\end{array} \right] , B = \left[ \begin{array} { l l l } 3 & 9 & 9 \\6 & 8 & 7 \\7 & 7 & 8\end{array} \right]$

Write the system of linear equations as a matrix equation,AX = B. $\left\{ \begin{array} { l } 2 x _ { 1 } + 3 x _ { 2 } = 8 \\4 x _ { 1 } + 5 x _ { 2 } = 16\end{array} \right.$

Evaluate the expression. $\frac { 3 } { 2 } \left[ \begin{array} { l l l } - 8 & - 1 & - 1\end{array} \right] + \left[ \begin{array} { l l l } - 9 & - 3 & 3\end{array} \right]$

Find A - B. $A = \left[ \begin{array} { c c } 4 & - 4 \\4 & 5 \\- 3 & 4\end{array} \right] , B = \left[ \begin{array} { c c } 3 & 8 \\- 3 & - 5 \\3 & 7\end{array} \right]$

If possible,find A². (Note: A²= AA. ) $A = \left[ \begin{array} { l l } 4 & 5 \\4 & 4\end{array} \right]$

Find 3A. $A = \left[ \begin{array} { c c c } - 2 & 5 & 0 \\5 & - 2 & 4 \\6 & 8 & - 1 \\0 & 6 & - 6 \\- 2 & - 1 & 0\end{array} \right]$

Find A - B. $A = \left[ \begin{array} { l l } 2 & - 2 \\4 & - 2\end{array} \right] , B = \left[ \begin{array} { c c } 4 & - 2 \\- 2 & 8\end{array} \right]$

Find x and y. $\left[ \begin{array} { c c } x & - 2 \\7 & y\end{array} \right] = \left[ \begin{array} { c c } - 20 & - 2 \\7 & 3\end{array} \right]$

Find 6A. $A = \left[ \begin{array} { l l } 1 & - 1 \\1 & - 1\end{array} \right]$

Find the product. $\left[ \begin{array} { c } 3 \\- 6 \\- 9\end{array} \right] \left[ \begin{array} { l l l } 5 & - 9 & - 5\end{array} \right]$