Question 1

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 2

Find A + B.   $$A = \left[ \begin{array} { c c c } 
- 1 & 2 & 0 \
5 & - 3 & 4 \
3 & 4 & - 1 \
0 & 6 & - 3 \
- 2 & - 1 & 0
\end{array} \right] , B = \left[ \begin{array} { c c c } 
- 1 & 8 & 4 \
3 & - 2 & - 4 \
9 & - 8 & - 2 \
4 & 2 & - 4 \
0 & 4 & - 4
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c c c } 
- 2 & 10 & 4 \
8 & - 5 & 0 \
12 & - 4 & - 3 \
4 & 8 & - 7 \
- 2 & 3 & - 4
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
0 & 10 & 4 \
8 & - 5 & 0 \
- 3 & - 4 & 12 \
4 & 8 & - 7 \
- 4 & 3 & - 2
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
2 & 10 & 4 \
0 & - 5 & 8 \
12 & - 4 & - 3 \
- 7 & 8 & 4 \
- 2 & 3 & - 4
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 6 & 10 & 4 \
8 & - 5 & 0 \
- 2 & - 4 & - 4 \
4 & 8 & - 7 \
12 & 3 & - 3
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c c } 
- 4 & 10 & 4 \
4 & - 5 & - 7 \
12 & - 4 & - 3 \
8 & 8 & 0 \
- 2 & 3 & - 4
\end{array} \right]$ 
A)  $\left[ \begin{array} { c c c } 
- 2 & 10 & 4 \
8 & - 5 & 0 \
12 & - 4 & - 3 \
4 & 8 & - 7 \
- 2 & 3 & - 4
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
0 & 10 & 4 \
8 & - 5 & 0 \
- 3 & - 4 & 12 \
4 & 8 & - 7 \
- 4 & 3 & - 2
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
2 & 10 & 4 \
0 & - 5 & 8 \
12 & - 4 & - 3 \
- 7 & 8 & 4 \
- 2 & 3 & - 4
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 6 & 10 & 4 \
8 & - 5 & 0 \
- 2 & - 4 & - 4 \
4 & 8 & - 7 \
12 & 3 & - 3
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c c } 
- 4 & 10 & 4 \
4 & - 5 & - 7 \
12 & - 4 & - 3 \
8 & 8 & 0 \
- 2 & 3 & - 4
\end{array} \right]$

Question 3

Use a system of equations to find the specified equation that passes through the points.Solve the system using matrices. Parabola: y = ax2 + bx + c

Accepted Answer

A)   $y = 2 x ^ { 2 } - x + 6$ 
B)   $y = - x ^ { 2 } + 6$ 
C)  $y = - x ^ { 2 } + x + 6$ 
D)   $y = + x ^ { 2 } + 6$ 
E)   $y = - x ^ { 2 } - x + 6$ 
A)   $y = 2 x ^ { 2 } - x + 6$ 
B)   $y = - x ^ { 2 } + 6$ 
C)  $y = - x ^ { 2 } + x + 6$ 
D)   $y = + x ^ { 2 } + 6$ 
E)   $y = - x ^ { 2 } - x + 6$

Question 4

If possible, find AB and state the order of the result.   $$A = \left[ \begin{array} { c c c } 
0 & 3 & 0 \
4 & 0 & 3 \
4 & 6 & - 3
\end{array} \right] , B = \left[ \begin{array} { c c } 
- 1 & 8 \
2 & - 2 \
10 & - 9
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c c } 
4 & - 6 \
5 & 26 \
- 22 & 47
\end{array} \right]$ Order: 3 × 2 
B)  $\left[ \begin{array} { c c } 
6 & - 6 \
26 & 5 \
- 22 & 47
\end{array} \right]$ Order: 3 × 2 
C)  $\left[ \begin{array} { c c } 
10 & 5 \
26 & - 6 \
- 22 & 47
\end{array} \right]$ Order: 3 × 2 
D)  $\left[ \begin{array} { c c } 
8 & - 6 \
- 22 & 5 \
26 & 47
\end{array} \right]$ Order: 3 × 2 
E)  $\left[ \begin{array} { c c } 
2 & - 6 \
26 & 5 \
47 & - 22
\end{array} \right]$ Order: 3 × 2 
A)  $\left[ \begin{array} { c c } 
4 & - 6 \
5 & 26 \
- 22 & 47
\end{array} \right]$ Order: 3 × 2 
B)  $\left[ \begin{array} { c c } 
6 & - 6 \
26 & 5 \
- 22 & 47
\end{array} \right]$ Order: 3 × 2 
C)  $\left[ \begin{array} { c c } 
10 & 5 \
26 & - 6 \
- 22 & 47
\end{array} \right]$ Order: 3 × 2 
D)  $\left[ \begin{array} { c c } 
8 & - 6 \
- 22 & 5 \
26 & 47
\end{array} \right]$ Order: 3 × 2 
E)  $\left[ \begin{array} { c c } 
2 & - 6 \
26 & 5 \
47 & - 22
\end{array} \right]$ Order: 3 × 2

Question 5

Determine the order of the matrix.  $$\left[ \begin{array} { l l l } 
4 & 3 & - 8
\end{array} \right]$$

Accepted Answer

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

Question 6

Write the system of linear equations represented by the augmented matrix.Then use back-substitution to solve.(Use variables x, y, and z.) $\left[\begin{array}{rrrrr}
1 & 2 & 5 & \vdots & 34 \
0 & 1 & 2 & \vdots & 14 \
0 & 0 & 1 & \vdots & 5
\end{array}\right]$

Accepted Answer

A) x = 35, y = -3, z = 1 
B) x = 4, y = 1, z = 5 
C) x = 1, y = -4, z = -5 
D) x = 1, y = 4, z = 5 
E) x = -4, y = 5, z = -1 
A) x = 35, y = -3, z = 1 
B) x = 4, y = 1, z = 5 
C) x = 1, y = -4, z = -5 
D) x = 1, y = 4, z = 5 
E) x = -4, y = 5, z = -1

Question 7

The currents in an electrical network are given by the solutions of the system $$\left\{ \begin{aligned}
I _ { 1 } + I _ { 2 } - I _ { 3 } & = 0 \
4 I _ { 1 } + 5 I _ { 3 } & = 23 \
6 I _ { 2 } + I _ { 3 } & = 9
\end{aligned} \right.$$ where I1, I2, and I3 are measured in amperes.Solve the system of equations using matrices.

Accepted Answer

A) I1 = 2, I2 = 3, I3 = 1 
B) I1 = 2, I2 = 1, I3 = 3 
C) I1 = 4, I2 = 4, I3 = 9 
D) I1 = 4, I2 = 4, I3 = 8 
E) no solution 
A) I1 = 2, I2 = 3, I3 = 1 
B) I1 = 2, I2 = 1, I3 = 3 
C) I1 = 4, I2 = 4, I3 = 9 
D) I1 = 4, I2 = 4, I3 = 8 
E) no solution

Question 8

Use a determinant to find y such that (8, -20), (16, y), and (20, -8) are collinear.

Accepted Answer

A) y = -12 
B) y = 20 
C) y = -44 
D) y = 4 
E) y = -8 
A) y = -12 
B) y = 20 
C) y = -44 
D) y = 4 
E) y = -8

Question 9

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 10

Find the determinant of $\left[ \begin{array} { c c c } 
6 & - 2 & 0 \
- 4 & 4 & 0 \
2 & 16 & 2
\end{array} \right]$ .

Accepted Answer

A) 32 
B) 0 
C) 24 
D) -56 
E) -64 
A) 32 
B) 0 
C) 24 
D) -56 
E) -64

Question 11

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

Accepted Answer

A)  $\left[ \begin{array} { c c } 
- 2 & - 2 \
16 & - 22
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
0 & - 22 \
16 & - 2
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
2 & - 2 \
- 22 & 16
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
- 4 & - 2 \
16 & - 22
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c } 
- 6 & - 2 \
16 & - 22
\end{array} \right]$ 
A)  $\left[ \begin{array} { c c } 
- 2 & - 2 \
16 & - 22
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
0 & - 22 \
16 & - 2
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
2 & - 2 \
- 22 & 16
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
- 4 & - 2 \
16 & - 22
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c } 
- 6 & - 2 \
16 & - 22
\end{array} \right]$

Question 12

Evaluate the determinant in which the entries are functions. 
 $\left| \begin{array} { c c } 
x & \ln x \
1 & \frac { 6 } { x }
\end{array} \right|$

Accepted Answer

A)  $x - \ln x$ 
B)  $6 x + \ln x$ 
C)  $6 + \ln x$ 
D)  $6 x - \ln x$ 
E)  $6 - \ln x$ 
A)  $x - \ln x$ 
B)  $6 x + \ln x$ 
C)  $6 + \ln x$ 
D)  $6 x - \ln x$ 
E)  $6 - \ln x$

Question 13

Determine a positive value for y such that a triangle with vertices $P ( 0,0 ) , Q ( 17,0 )$ , and $R ( 17 , y )$ has an area of 17 square units.

Accepted Answer

A) 8 
B) 2 
C) 9 
D) 11 
E) 1 
A) 8 
B) 2 
C) 9 
D) 11 
E) 1

Question 14

Use Cramer's Rule to solve (if possible) the system of equations.   $$\left\{ \begin{array} { r } 
- 7 x + 11 y = 20 \
3 x - 9 y = 0
\end{array} \right.$$

Accepted Answer

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

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

Find the inverse of the matrix (if it exists).   $\left[ \begin{array} { c c c } 
4 & 0 & 0 \
6 & 0 & 0 \
5 & 11 & 11
\end{array} \right]$

Accepted Answer

A)   $\left[ \begin{array} { c c c } 
- 4 & 0 & 0 \
- 6 & 0 & 0 \
5 & 4 & 11
\end{array} \right]$ 
B)    $\left[ \begin{array} { c c c } 
- 4 & 0 & 0 \
- 6 & 0 & 0 \
- 5 & - 4 & - 11
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c c } 
4 & 0 & 0 \
6 & 0 & 0 \
5 & 4 & 11
\end{array} \right]$ 
D)   $\left[ \begin{array} { c c c } 
- 4 & 0 & 0 \
- 6 & 0 & 0 \
- 5 & 4 & 11
\end{array} \right]$ 
E) Does not exist 
A)   $\left[ \begin{array} { c c c } 
- 4 & 0 & 0 \
- 6 & 0 & 0 \
5 & 4 & 11
\end{array} \right]$ 
B)    $\left[ \begin{array} { c c c } 
- 4 & 0 & 0 \
- 6 & 0 & 0 \
- 5 & - 4 & - 11
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c c } 
4 & 0 & 0 \
6 & 0 & 0 \
5 & 4 & 11
\end{array} \right]$ 
D)   $\left[ \begin{array} { c c c } 
- 4 & 0 & 0 \
- 6 & 0 & 0 \
- 5 & 4 & 11
\end{array} \right]$ 
E) Does not exist

Question 17

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 18

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)  $\left[ \begin{array} { c c } 
- 17 & - 17 \
19 & 8 \
- 11 & - 6
\end{array} \right]$

Question 19

Use Cramer's Rule to solve the following system of linear equations:   $$\left\{ \begin{aligned}
3 x - 6 z & = 8 \
- 6 y + 13 z & = 3 \
6 x + 15 z & = 0
\end{aligned} \right.$$

Accepted Answer

A)  x = $\frac { 40 } { 27 }$ , y = $- \frac { 16 } { 27 }$ , z = $- \frac { 289 } { 162 }$ 
B)  x = $\frac { 40 } { 27 }$ , y = $- \frac { 289 } { 162 }$ , z = $- \frac { 16 } { 27 }$ 
C)  x = $\frac { 27 } { 40 }$ , y = $- \frac { 162 } { 289 }$ , z = $- \frac { 27 } { 16 }$ 
D)  x = $\frac { 27 } { 40 }$ , y = $- \frac { 289 } { 162 }$ , z = $- \frac { 16 } { 27 }$ 
E)  x = $\frac { 40 } { 27 }$ , y = $- \frac { 162 } { 289 }$ , z = $- \frac { 16 } { 27 }$ 
A)  x = $\frac { 40 } { 27 }$ , y = $- \frac { 16 } { 27 }$ , z = $- \frac { 289 } { 162 }$ 
B)  x = $\frac { 40 } { 27 }$ , y = $- \frac { 289 } { 162 }$ , z = $- \frac { 16 } { 27 }$ 
C)  x = $\frac { 27 } { 40 }$ , y = $- \frac { 162 } { 289 }$ , z = $- \frac { 27 } { 16 }$ 
D)  x = $\frac { 27 } { 40 }$ , y = $- \frac { 289 } { 162 }$ , z = $- \frac { 16 } { 27 }$ 
E)  x = $\frac { 40 } { 27 }$ , y = $- \frac { 162 } { 289 }$ , z = $- \frac { 16 } { 27 }$

Question 20

Find x and y.   $$\left[ \begin{array} { c c } 
- 5 & x \
y & 8
\end{array} \right] = \left[ \begin{array} { c c } 
- 5 & 14 \
27 & 8
\end{array} \right]$$

Accepted Answer

A)  $x = 14 , y = 27$ 
B)   $x = \frac { 1 } { 14 } , y = \frac { 1 } { 27 }$ 
C)  $x = \frac { 1 } { 14 } , y = - \frac { 1 } { 27 }$ 
D)  $x = - 14 , y = - 27$ 
E)  $x = - \frac { 1 } { 14 } , y = \frac { 1 } { 27 }$ 
A)  $x = 14 , y = 27$ 
B)   $x = \frac { 1 } { 14 } , y = \frac { 1 } { 27 }$ 
C)  $x = \frac { 1 } { 14 } , y = - \frac { 1 } { 27 }$ 
D)  $x = - 14 , y = - 27$ 
E)  $x = - \frac { 1 } { 14 } , y = \frac { 1 } { 27 }$

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

Find A + B. $A = \left[ \begin{array} { c c c } - 1 & 2 & 0 \\5 & - 3 & 4 \\3 & 4 & - 1 \\0 & 6 & - 3 \\- 2 & - 1 & 0\end{array} \right] , B = \left[ \begin{array} { c c c } - 1 & 8 & 4 \\3 & - 2 & - 4 \\9 & - 8 & - 2 \\4 & 2 & - 4 \\0 & 4 & - 4\end{array} \right]$

Use a system of equations to find the specified equation that passes through the points.Solve the system using matrices. Parabola: y = ax² + bx + c

If possible, find AB and state the order of the result. $A = \left[ \begin{array} { c c c } 0 & 3 & 0 \\4 & 0 & 3 \\4 & 6 & - 3\end{array} \right] , B = \left[ \begin{array} { c c } - 1 & 8 \\2 & - 2 \\10 & - 9\end{array} \right]$

Determine the order of the matrix. $\left[ \begin{array} { l l l } 4 & 3 & - 8\end{array} \right]$

Write the system of linear equations represented by the augmented matrix.Then use back-substitution to solve.(Use variables x, y, and z.) $\left[\begin{array}{rrrrr}1 & 2 & 5 & \vdots & 34 \\0 & 1 & 2 & \vdots & 14 \\0 & 0 & 1 & \vdots & 5\end{array}\right]$

Use a determinant to find y such that (8, -20), (16, y), and (20, -8) are collinear.

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 determinant of $\left[ \begin{array} { c c c } 6 & - 2 & 0 \\- 4 & 4 & 0 \\2 & 16 & 2\end{array} \right]$ .

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

Evaluate the determinant in which the entries are functions. $\left| \begin{array} { c c } x & \ln x \\1 & \frac { 6 } { x }\end{array} \right|$

Determine a positive value for y such that a triangle with vertices $P ( 0,0 ) , Q ( 17,0 )$ , and $R ( 17 , y )$ has an area of 17 square units.

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { r } - 7 x + 11 y = 20 \\3 x - 9 y = 0\end{array} \right.$

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

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { c c c } 4 & 0 & 0 \\6 & 0 & 0 \\5 & 11 & 11\end{array} \right]$

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

Use Cramer's Rule to solve the following system of linear equations: $\left\{ \begin{aligned}3 x - 6 z & = 8 \\- 6 y + 13 z & = 3 \\6 x + 15 z & = 0\end{aligned} \right.$

Find x and y. $\left[ \begin{array} { c c } - 5 & x \\y & 8\end{array} \right] = \left[ \begin{array} { c c } - 5 & 14 \\27 & 8\end{array} \right]$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Exam 8: Matrices and Determinants

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​]

Use a system of equations to find the specified equation that passes through the points.Solve the system using matrices. Parabola: y = ax2 + bx + c

Determine the order of the matrix. [43−8]\left[ \begin{array} { l l l } 4 & 3 & - 8\end{array} \right][4​3​−8​]

Use a determinant to find y such that (8, -20), (16, y), and (20, -8) are collinear.

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 determinant of [6−20−4402162]\left[ \begin{array} { c c c } 6 & - 2 & 0 \\- 4 & 4 & 0 \\2 & 16 & 2\end{array} \right]​6−42​−2416​002​​ .

Find 3A - 2B. 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​]

Evaluate the determinant in which the entries are functions. ∣xln⁡x16x∣\left| \begin{array} { c c } x & \ln x \\1 & \frac { 6 } { x }\end{array} \right|​x1​lnxx6​​​

Determine a positive value for y such that a triangle with vertices P(0,0),Q(17,0)P ( 0,0 ) , Q ( 17,0 )P(0,0),Q(17,0) , and R(17,y)R ( 17 , y )R(17,y) has an area of 17 square units.

Use Cramer's Rule to solve (if possible) the system of equations. {−7x+11y=203x−9y=0\left\{ \begin{array} { r } - 7 x + 11 y = 20 \\3 x - 9 y = 0\end{array} \right.{−7x+11y=203x−9y=0​

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​

Find the inverse of the matrix (if it exists). [40060051111]\left[ \begin{array} { c c c } 4 & 0 & 0 \\6 & 0 & 0 \\5 & 11 & 11\end{array} \right]​465​0011​0011​​

Use Cramer's Rule to solve the following system of linear equations: {3x−6z=8−6y+13z=36x+15z=0\left\{ \begin{aligned}3 x - 6 z & = 8 \\- 6 y + 13 z & = 3 \\6 x + 15 z & = 0\end{aligned} \right.⎩⎨⎧​3x−6z−6y+13z6x+15z​=8=3=0​

Find x and y. [−5xy8]=[−514278]\left[ \begin{array} { c c } - 5 & x \\y & 8\end{array} \right] = \left[ \begin{array} { c c } - 5 & 14 \\27 & 8\end{array} \right][−5y​x8​]=[−527​148​]

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

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

Use a system of equations to find the specified equation that passes through the points.Solve the system using matrices. Parabola: y = ax² + bx + c

Determine the order of the matrix. $\left[ \begin{array} { l l l } 4 & 3 & - 8\end{array} \right]$

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 determinant of $\left[ \begin{array} { c c c } 6 & - 2 & 0 \\- 4 & 4 & 0 \\2 & 16 & 2\end{array} \right]$ .

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

Evaluate the determinant in which the entries are functions. $\left| \begin{array} { c c } x & \ln x \\1 & \frac { 6 } { x }\end{array} \right|$

Determine a positive value for y such that a triangle with vertices $P ( 0,0 ) , Q ( 17,0 )$ , and $R ( 17 , y )$ has an area of 17 square units.

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { r } - 7 x + 11 y = 20 \\3 x - 9 y = 0\end{array} \right.$

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

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { c c c } 4 & 0 & 0 \\6 & 0 & 0 \\5 & 11 & 11\end{array} \right]$

Use Cramer's Rule to solve the following system of linear equations: $\left\{ \begin{aligned}3 x - 6 z & = 8 \\- 6 y + 13 z & = 3 \\6 x + 15 z & = 0\end{aligned} \right.$

Find x and y. $\left[ \begin{array} { c c } - 5 & x \\y & 8\end{array} \right] = \left[ \begin{array} { c c } - 5 & 14 \\27 & 8\end{array} \right]$