Question 1

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 2

Write the system of linear equations represented by the augmented matrix.(Use variables x, y, z, and w.) $\left[ \begin{array} { c c c c c } 
-1&0&0&3 & \vdots &  2 \
-7&4&0&0 & \vdots & 8 \
0&5&3&-5 & \vdots & - 8\
0&0&-1&-9& \vdots & 9\
\end{array} \right]$

Accepted Answer

A)  $\left\{ \begin{array} { r l r } 
- x + 3 w & = 2 \
- 7 x + 4 y & = 8 \
5 y + 3 z - 5 w & = - 8 \
- y - 9 z & = 9
\end{array} \right.$ 
B)  $\left\{ \begin{array} { r l r } 
- x + 3 z & = 2 \
- 7 x + 4 z & = 8 \
5 y + 3 z - 5 w & = - 8 \
- z - 9 w & = 9
\end{array} \right.$ 
C)  $\left\{ \begin{array} { r l r } 
- x + 3 y & = & 2 \
- 7 x + 4 y & = & 8 \
5 x + 3 y - 5 z & = - 8 \
- x - 9 y & = 9
\end{array} \right.$ 
D)  $\left\{ \begin{aligned}
- x + 3 w & = 2 \
- 7 x + 4 y & = 8 \
5 y + 3 z - 5 w & = - 8 \
- z - 9 w & = 9
\end{aligned} \right.$ 
E)  $\left\{ \begin{array} { r l r } 
- x + 3 z & = 2 \
- 7 x + 4 y & = 8 \
5 y + 3 z - 5 w & = - 8 \
- z - 9 w & = 9
\end{array} \right.$ 
A)  $\left\{ \begin{array} { r l r } 
- x + 3 w & = 2 \
- 7 x + 4 y & = 8 \
5 y + 3 z - 5 w & = - 8 \
- y - 9 z & = 9
\end{array} \right.$ 
B)  $\left\{ \begin{array} { r l r } 
- x + 3 z & = 2 \
- 7 x + 4 z & = 8 \
5 y + 3 z - 5 w & = - 8 \
- z - 9 w & = 9
\end{array} \right.$ 
C)  $\left\{ \begin{array} { r l r } 
- x + 3 y & = & 2 \
- 7 x + 4 y & = & 8 \
5 x + 3 y - 5 z & = - 8 \
- x - 9 y & = 9
\end{array} \right.$ 
D)  $\left\{ \begin{aligned}
- x + 3 w & = 2 \
- 7 x + 4 y & = 8 \
5 y + 3 z - 5 w & = - 8 \
- z - 9 w & = 9
\end{aligned} \right.$ 
E)  $\left\{ \begin{array} { r l r } 
- x + 3 z & = 2 \
- 7 x + 4 y & = 8 \
5 y + 3 z - 5 w & = - 8 \
- z - 9 w & = 9
\end{array} \right.$

Question 3

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 4

Select the order for the following matrix.   $\left[ \begin{array} { l l l l } 
5 & - 4 & 3 & 3
\end{array} \right]$

Accepted Answer

A) 2 × 2 
B) 4 × 4 
C) 3 × 1 
D) 4 × 1 
E) 1 × 4 
A) 2 × 2 
B) 4 × 4 
C) 3 × 1 
D) 4 × 1 
E) 1 × 4

Question 5

You inherited a triangular piece of property after your Uncle Izzy passed away.You want to know the size of land, so you "step it off" to estimate the square footage.From the southernmost vertex A, you travel north 300 feet then west 220 feet (for vertex C), and from the southernmost vertex A, you travel 420 feet north then 50 feet west (for vertex B).Use a graphing utility to approximate the number of square feet of land that you have inherited.

Accepted Answer

A) 3,000 square feet 
B) 38,700 square feet 
C) 53,700 square feet 
D) 43,500 square feet 
E) 22,500 square feet 
A) 3,000 square feet 
B) 38,700 square feet 
C) 53,700 square feet 
D) 43,500 square feet 
E) 22,500 square feet

Question 6

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 7

Use a determinant to find an equation of the line passing through the points (-5, -1) and (2, 4). ​

Accepted Answer

A) 5x + 7y - 18 = 0 
B) -5x - 7y - 18 = 0 
C) -5x + 7y + 18 = 0 
D) 5x - 7y - 18 = 0 
E) -5x + 7y - 18 = 0 
A) 5x + 7y - 18 = 0 
B) -5x - 7y - 18 = 0 
C) -5x + 7y + 18 = 0 
D) 5x - 7y - 18 = 0 
E) -5x + 7y - 18 = 0

Question 8

Solve for x. 
 $\left| \begin{array} { l l } 
x & 3 \
1 & x
\end{array} \right| = 6$

Accepted Answer

A)  $x = 3$ 
B)  $x = \pm 2$ 
C)  $x = \pm 3$ 
D)  $x = - 4$ 
E)  $x = \pm 4$ 
A)  $x = 3$ 
B)  $x = \pm 2$ 
C)  $x = \pm 3$ 
D)  $x = - 4$ 
E)  $x = \pm 4$

Question 9

Find the system of linear equations represented by the augmented matrix.Then use back substitution to solve.(Use variables x, y, z, and w if applicable.)   $$\left[ \begin{array} { c c c c } 
1 & 7 & \vdots & 0 \
0 & 1 & \vdots & - 1
\end{array} \right]$$

Accepted Answer

A)   $\left\{ \begin{array} { r } 
x + 7 y = 0 \
y = 0
\end{array} \right.$ (7, 0) 
B)   $\left\{ \begin{aligned}
x + 7 y & = 0 \
x & = - 1
\end{aligned} \right.$  (-1, 7) 
C)   $\left\{ \begin{aligned}
x + 7 y & = 0 \
y & = - 1
\end{aligned} \right.$  (7, -1) 
D)   $\left\{ \begin{array} { r } 
x + 7 y = 0 \
y = 1
\end{array} \right.$  (7, 1) 
E)   $\left\{ \begin{aligned}
x + 7 y & = 0 \
y & = - 1
\end{aligned} \right.$  (7, 1) 
A)   $\left\{ \begin{array} { r } 
x + 7 y = 0 \
y = 0
\end{array} \right.$ (7, 0) 
B)   $\left\{ \begin{aligned}
x + 7 y & = 0 \
x & = - 1
\end{aligned} \right.$  (-1, 7) 
C)   $\left\{ \begin{aligned}
x + 7 y & = 0 \
y & = - 1
\end{aligned} \right.$  (7, -1) 
D)   $\left\{ \begin{array} { r } 
x + 7 y = 0 \
y = 1
\end{array} \right.$  (7, 1) 
E)   $\left\{ \begin{aligned}
x + 7 y & = 0 \
y & = - 1
\end{aligned} \right.$  (7, 1)

Question 10

Find the inverse of the matrix $\left[ \begin{array} { c c } 
- 1 & 4 \
3 & - 2
\end{array} \right]$ (if it exists).

Accepted Answer

A)   $- \frac { 1 } { - 10 } \left[ \begin{array} { l l } 
- 2 & - 4 \
- 3 & - 1
\end{array} \right]$ 
B)  $- \frac { 1 } { - 10 } \left[ \begin{array} { c c } 
- 1 & 4 \
3 & - 2
\end{array} \right]$ 
C)   $\frac { 1 } { - 10 } \left[ \begin{array} { l l } 
- 2 & - 4 \
- 3 & - 1
\end{array} \right]$ 
D)   $\frac { 1 } { - 10 } \left[ \begin{array} { l l } 
2 & 4 \
3 & 1
\end{array} \right]$ 
E)  does not exist 
A)   $- \frac { 1 } { - 10 } \left[ \begin{array} { l l } 
- 2 & - 4 \
- 3 & - 1
\end{array} \right]$ 
B)  $- \frac { 1 } { - 10 } \left[ \begin{array} { c c } 
- 1 & 4 \
3 & - 2
\end{array} \right]$ 
C)   $\frac { 1 } { - 10 } \left[ \begin{array} { l l } 
- 2 & - 4 \
- 3 & - 1
\end{array} \right]$ 
D)   $\frac { 1 } { - 10 } \left[ \begin{array} { l l } 
2 & 4 \
3 & 1
\end{array} \right]$ 
E)  does not exist

Question 11

Use Cramer's Rule to solve the following system of linear equations: $$\left\{ \begin{aligned}
14 x - 21 y + 14 z & = 1 \
- 21 x + 14 y + 7 z & = 5 \
28 x + 7 y - 21 z & = 5
\end{aligned} \right.$$

Accepted Answer

A)  $x = \frac { 5 } { 7 } ; y = \frac { 55 } { 49 } ; z = \frac { 95 } { 147 }$ 
B)  $x = \frac { 3 } { 7 } ; y = \frac { 45 } { 98 } ; z = \frac { 12 } { 49 }$ 
C)  $x = \frac { 11 } { 14 } ; y = \frac { 115 } { 98 } ; z = \frac { 106 } { 147 }$ 
D)  $x = \frac { 11 } { 21 } ; y = \frac { 115 } { 147 } ; z = \frac { 106 } { 147 }$ 
E)  $x = \frac { 3 } { 7 } ; y = \frac { 75 } { 98 } ; z = \frac { 27 } { 49 }$ 
A)  $x = \frac { 5 } { 7 } ; y = \frac { 55 } { 49 } ; z = \frac { 95 } { 147 }$ 
B)  $x = \frac { 3 } { 7 } ; y = \frac { 45 } { 98 } ; z = \frac { 12 } { 49 }$ 
C)  $x = \frac { 11 } { 14 } ; y = \frac { 115 } { 98 } ; z = \frac { 106 } { 147 }$ 
D)  $x = \frac { 11 } { 21 } ; y = \frac { 115 } { 147 } ; z = \frac { 106 } { 147 }$ 
E)  $x = \frac { 3 } { 7 } ; y = \frac { 75 } { 98 } ; z = \frac { 27 } { 49 }$

Question 12

Perform the sequence of row operations on the following matrix. Add -3 times R1 to R2.
  $\left[ \begin{array} { c c c } 
1 & 3 & 4 \
3 & - 1 & - 5 \
4 & 1 & - 1
\end{array} \right]$

Accepted Answer

A)   $\left[ \begin{array} { c c c } 
1 & 3 & 4 \
0 & 4 & - 17 \
4 & 1 & - 1
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
1 & 3 & 4 \
0 & - 10 & - 17 \
4 & 1 & - 1
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
1 & 3 & 4 \
0 & - 10 & 3 \
4 & 1 & - 1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 1 & 3 & 4 \
0 & - 10 & - 17 \
4 & - 1 & - 1
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c c } 
1 & 3 & 4 \
0 & 1 & - 17 \
4 & 1 & - 1
\end{array} \right]$ 
A)   $\left[ \begin{array} { c c c } 
1 & 3 & 4 \
0 & 4 & - 17 \
4 & 1 & - 1
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
1 & 3 & 4 \
0 & - 10 & - 17 \
4 & 1 & - 1
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
1 & 3 & 4 \
0 & - 10 & 3 \
4 & 1 & - 1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 1 & 3 & 4 \
0 & - 10 & - 17 \
4 & - 1 & - 1
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c c } 
1 & 3 & 4 \
0 & 1 & - 17 \
4 & 1 & - 1
\end{array} \right]$

Question 13

Find the determinant of the matrix.   $[ - 15 ]$

Accepted Answer

A)  $15$ 
B)  $- \frac { 1 } { 15 }$ 
C)  $- 15$ 
D) 0 
E)  $\frac { 1 } { 15 }$ 
A)  $15$ 
B)  $- \frac { 1 } { 15 }$ 
C)  $- 15$ 
D) 0 
E)  $\frac { 1 } { 15 }$

Question 14

Find the determinant of the matrix.   $[ 6 ]$

Accepted Answer

A) 0 
B)  $- 6$ 
C)  $- \frac { 1 } { 6 }$ 
D)  $6$ 
E)  $\frac { 1 } { 6 }$ 
A) 0 
B)  $- 6$ 
C)  $- \frac { 1 } { 6 }$ 
D)  $6$ 
E)  $\frac { 1 } { 6 }$

Question 15

Find the determinant of the matrix.   $\left[ \begin{array} { c c } 
10 & 3 \
0 & 11
\end{array} \right]$

Accepted Answer

A) 111 
B) 110 
C)  $- 111$ 
D)  $- 110$ 
E)  $109$ 
A) 111 
B) 110 
C)  $- 111$ 
D)  $- 110$ 
E)  $109$

Question 16

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 17

Write the system of linear equations represented by the augmented matrix.(Use variables x, y, z, and w.) $\left[ \begin{array} { r r r r r } 
1&0&0&4 &  \vdots&-5 \
3&-7&0&0 & \vdots & 3 \
0&-4&8&-1& \vdots &-6\
0&0&3&7 & \vdots &7\
\end{array} \right]$

Accepted Answer

A)  $\left\{ \begin{array} { r l r } 
x + 4 y & = & - 5 \
3 x - 7 y & = & 3 \
- 4 x + 8 y - z & = & - 6 \
3 x + 7 y & = & 7
\end{array} \right.$ 
B)  $\left\{ \begin{array} { r l r } 
x + 4 z & = - 5 \
3 x - 7 y & = 3 \
- 4 y + 8 z - w & = - 6 \
3 z z + 7 w & = 7
\end{array} \right.$ 
C)  $\left\{ \begin{array} { r l r } 
x + 4 z & = - 5 \
3 x - 7 z & = 3 \
- 4 y + 8 z - w & = - 6 \
3 z z + 7 w & = 7
\end{array} \right.$ 
D)  $\left\{ \begin{array} { r l r } 
x - 4 w & = - 5 \
3 x - 7 y & = 3 \
- 4 y + 8 z - w & = - 6 \
3 y + 7 z & = 7
\end{array} \right.$ 
E)  $\left\{ \begin{aligned}
x + 4 w & = - 5 \
3 x - 7 y & = 3 \
- 4 y + 8 z - w & = - 6 \
3 z + 7 w & = 7
\end{aligned} \right.$ 
A)  $\left\{ \begin{array} { r l r } 
x + 4 y & = & - 5 \
3 x - 7 y & = & 3 \
- 4 x + 8 y - z & = & - 6 \
3 x + 7 y & = & 7
\end{array} \right.$ 
B)  $\left\{ \begin{array} { r l r } 
x + 4 z & = - 5 \
3 x - 7 y & = 3 \
- 4 y + 8 z - w & = - 6 \
3 z z + 7 w & = 7
\end{array} \right.$ 
C)  $\left\{ \begin{array} { r l r } 
x + 4 z & = - 5 \
3 x - 7 z & = 3 \
- 4 y + 8 z - w & = - 6 \
3 z z + 7 w & = 7
\end{array} \right.$ 
D)  $\left\{ \begin{array} { r l r } 
x - 4 w & = - 5 \
3 x - 7 y & = 3 \
- 4 y + 8 z - w & = - 6 \
3 y + 7 z & = 7
\end{array} \right.$ 
E)  $\left\{ \begin{aligned}
x + 4 w & = - 5 \
3 x - 7 y & = 3 \
- 4 y + 8 z - w & = - 6 \
3 z + 7 w & = 7
\end{aligned} \right.$

Question 18

Find $| A B |$ . 
 $A = \left[ \begin{array} { c c } 
- 2 & 0 \
0 & 5
\end{array} \right] , B = \left[ \begin{array} { c c } 
4 & 0 \
0 & - 2
\end{array} \right]$

Accepted Answer

A) 80 
B) 78 
C) 79 
D) 82 
E) 81 
A) 80 
B) 78 
C) 79 
D) 82 
E) 81

Question 19

Determine whether the following matrix is in row-echelon form.If it is, determine if it is also in reduced row-echelon form.  
  $$\left[ \begin{array} { l l l l } 
1 & 0 & 1 & 0 \
0 & 1 & 0 & 3 \
0 & 0 & 1 & 0
\end{array} \right]$$

Accepted Answer

A) Row-echelon form 
B) Not in row-echelon form 
C) Reduced row-echelon form 
A) Row-echelon form 
B) Not in row-echelon form 
C) Reduced row-echelon form

Question 20

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)   $\left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 } \
x _ { 4 }
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 24 } { 5 } \
\frac { - 8 } { 5 } \
\frac { - 16 } { 5 } \
\frac { 16 } { 5 }
\end{array} \right]$

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

Write the system of linear equations represented by the augmented matrix.(Use variables x, y, z, and w.) $\left[ \begin{array} { c c c c c } -1&0&0&3 & \vdots & 2 \\-7&4&0&0 & \vdots & 8 \\0&5&3&-5 & \vdots & - 8\\0&0&-1&-9& \vdots & 9\\\end{array} \right]$

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). $\left[ \begin{array} { l l } 2 & 3 \\1 & 5\end{array} \right]$

Select the order for the following matrix. $\left[ \begin{array} { l l l l } 5 & - 4 & 3 & 3\end{array} \right]$

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

Use a determinant to find an equation of the line passing through the points (-5, -1) and (2, 4).

Solve for x. $\left| \begin{array} { l l } x & 3 \\1 & x\end{array} \right| = 6$

Find the system of linear equations represented by the augmented matrix.Then use back substitution to solve.(Use variables x, y, z, and w if applicable.) $\left[ \begin{array} { c c c c } 1 & 7 & \vdots & 0 \\0 & 1 & \vdots & - 1\end{array} \right]$

Find the inverse of the matrix $\left[ \begin{array} { c c } - 1 & 4 \\3 & - 2\end{array} \right]$ (if it exists).

Use Cramer's Rule to solve the following system of linear equations: $\left\{ \begin{aligned}14 x - 21 y + 14 z & = 1 \\- 21 x + 14 y + 7 z & = 5 \\28 x + 7 y - 21 z & = 5\end{aligned} \right.$

Perform the sequence of row operations on the following matrix. Add -3 times R₁ to R₂. $\left[ \begin{array} { c c c } 1 & 3 & 4 \\3 & - 1 & - 5 \\4 & 1 & - 1\end{array} \right]$

Find the determinant of the matrix. $[ - 15 ]$

Find the determinant of the matrix. $[ 6 ]$

Find the determinant of the matrix. $\left[ \begin{array} { c c } 10 & 3 \\0 & 11\end{array} \right]$

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

Write the system of linear equations represented by the augmented matrix.(Use variables x, y, z, and w.) $\left[ \begin{array} { r r r r r } 1&0&0&4 & \vdots&-5 \\3&-7&0&0 & \vdots & 3 \\0&-4&8&-1& \vdots &-6\\0&0&3&7 & \vdots &7\\\end{array} \right]$

Find $| A B |$ . $A = \left[ \begin{array} { c c } - 2 & 0 \\0 & 5\end{array} \right] , B = \left[ \begin{array} { c c } 4 & 0 \\0 & - 2\end{array} \right]$

Determine whether the following matrix is in row-echelon form.If it is, determine if it is also in reduced row-echelon form. $\left[ \begin{array} { l l l l } 1 & 0 & 1 & 0 \\0 & 1 & 0 & 3 \\0 & 0 & 1 & 0\end{array} \right]$

Functions and Their Graphs

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Exam 8: Matrices and Determinants

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​

Select the order for the following matrix. [5−433]\left[ \begin{array} { l l l l } 5 & - 4 & 3 & 3\end{array} \right][5​−4​3​3​]