Question 1

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

Accepted Answer

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

Question 2

Fill in the blank using elementary row operations to form a row-equivalent matrix. $\left[ \begin{array} { r r r } 
2 & 1 & - 2 \
- 10 & - 1 & 9
\end{array} \right]$ $\left[ \begin{array} { l l l } 
2 & 1 & - 2 \
0 & \square & - 1
\end{array} \right]$

Accepted Answer

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

Question 3

Find A + B.
​ $$A = \left[ \begin{array} { l l l } 
5 & 5 & 4 \
9 & 9 & 2
\end{array} \right] , B = \left[ \begin{array} { l l l } 
3 & 7 & 8 \
7 & 6 & 4
\end{array} \right]$$

Accepted Answer

The answer of Find A + B.
​ \[A = \left[...

Question 4

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

Accepted Answer

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

Question 5

An augmented matrix that represents a system of linear equations (in variables x, y, z and w if applicable) has been reduced using Gauss-Jordan elimination.Find the solution represented by the augmented matrix.   $\left[ \begin{array} { c c c c } 
1&0&0 &\vdots& 8 \
0&1&0 &\vdots& -6 \
0 & 0 & 1 &\vdots&0\
\end{array} \right]$

Accepted Answer

A)  (0, 8, -6) 
B) (8, 0, -6) 
C) (8, -6, 0) 
D) (-8, -6, 0) 
E) (8, 6, 0) 
A)  (0, 8, -6) 
B) (8, 0, -6) 
C) (8, -6, 0) 
D) (-8, -6, 0) 
E) (8, 6, 0)

Question 6

Write the matrix in reduced row-echelon form. $$\left[ \begin{array} { r r r r } 
9 & - 6 & - 3 & - 33 \
8 & - 1 & 1 & - 47 \
- 7 & - 7 & 6 & - 8
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { l l l l } 
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l l r } 
1 & 0 & 0 & - 5 \
0 & 1 & 0 & - 6 \
0 & 0 & 1 & 1
\end{array} \right]$ 
C)  $\left[ \begin{array} { r r r r } 
1 & 0 & 0 & - 6 \
0 & 1 & 0 & 2 \
0 & 0 & 1 & - 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { r r r r } 
1 & 0 & 0 & - 5 \
0 & 1 & 0 & 1 \
0 & 0 & 1 & - 6
\end{array} \right]$ 
E)  $\left[ \begin{array} { l l l l } 
1 & 0 & 0 & 1 \
0 & 1 & 0 & 1 \
0 & 0 & 1 & 1
\end{array} \right]$ 
A)  $\left[ \begin{array} { l l l l } 
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l l r } 
1 & 0 & 0 & - 5 \
0 & 1 & 0 & - 6 \
0 & 0 & 1 & 1
\end{array} \right]$ 
C)  $\left[ \begin{array} { r r r r } 
1 & 0 & 0 & - 6 \
0 & 1 & 0 & 2 \
0 & 0 & 1 & - 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { r r r r } 
1 & 0 & 0 & - 5 \
0 & 1 & 0 & 1 \
0 & 0 & 1 & - 6
\end{array} \right]$ 
E)  $\left[ \begin{array} { l l l l } 
1 & 0 & 0 & 1 \
0 & 1 & 0 & 1 \
0 & 0 & 1 & 1
\end{array} \right]$

Question 7

Use a determinant to find y such that $( 6 , - 15 ) , ( 12 , y )$ , and $( 15 , - 6 )$ are collinear.

Accepted Answer

A)  $y = - 33$ 
B)  $y = 15$ 
C)  $y = 3$ 
D)  $y = - 6$ 
E)  $y = - 9$ 
A)  $y = - 33$ 
B)  $y = 15$ 
C)  $y = 3$ 
D)  $y = - 6$ 
E)  $y = - 9$

Question 8

Find the product.   $$\left[ \begin{array} { l l } 
9 & 3 \
7 & 8
\end{array} \right] \left[ \begin{array} { l l } 
8 & 7 \
8 & 4
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c c } 
72 & 75 \
120 & 81
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 
72 & 21 \
56 & 32
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
96 & 75 \
56 & 32
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
96 & 75 \
120 & 81
\end{array} \right]$ 
E) cannot multiply 
A)  $\left[ \begin{array} { c c } 
72 & 75 \
120 & 81
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 
72 & 21 \
56 & 32
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
96 & 75 \
56 & 32
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
96 & 75 \
120 & 81
\end{array} \right]$ 
E) cannot multiply

Question 9

Find the determinant of the matrix by the method of expansion by cofactors.Expand using the column 2.   $\left[ \begin{array} { c c c } 
- 3 & 2 & 1 \
4 & 5 & 6 \
2 & - 3 & 1
\end{array} \right]$

Accepted Answer

A) -75 
B) 75 
C) -73 
D) -74 
E) -76 
A) -75 
B) 75 
C) -73 
D) -74 
E) -76

Question 10

Use Cramer's Rule to solve (if possible) the system of equations.  
  $$\left\{ \begin{array} { r } 
5 x - 4 y + z = - 12 \
- x + 2 y - 2 z = 12 \
3 x + y + z = - 2
\end{array} \right.$$

Accepted Answer

A) (0, 2, -4) 
B) (0, -2, 4) 
C) (0, -2, -4) 
D) (0, 2, 0) 
E) (0, 2, 4) 
A) (0, 2, -4) 
B) (0, -2, 4) 
C) (0, -2, -4) 
D) (0, 2, 0) 
E) (0, 2, 4)

Question 11

Use a determinant to determine whether the points (-3, -9), (-5, -11) and (0, -7) are collinear.

Accepted Answer

A)  $\left| \begin{array} { c c c } 
- 3 & - 9 & 1 \
- 5 & - 11 & 1 \
0 & - 7 & 1
\end{array} \right| = 2 \text {; therefore, the points are not collinear }$ 
B)  $\left| \begin{array} { c c c } 
- 3 & - 9 & 1 \
- 5 & - 11 & 1 \
0 & - 7 & 1
\end{array} \right| = 0 \text {; therefore, the points are collinear }$ 
A)  $\left| \begin{array} { c c c } 
- 3 & - 9 & 1 \
- 5 & - 11 & 1 \
0 & - 7 & 1
\end{array} \right| = 2 \text {; therefore, the points are not collinear }$ 
B)  $\left| \begin{array} { c c c } 
- 3 & - 9 & 1 \
- 5 & - 11 & 1 \
0 & - 7 & 1
\end{array} \right| = 0 \text {; therefore, the points are collinear }$

Question 12

Find the system of linear equations represented by the augmented matrix.Then use back substitution to solve.(Use variables x, y, z and if applicable.)  $\left[ \begin{array} { c c c c } 
1&-1&6 &\vdots& 8 \
0&1&-1 &\vdots& 6 \
0 & 0 & 1 &\vdots&-6\
\end{array} \right]$

Accepted Answer

A)   $\left\{ \begin{aligned}
x - y + 6 z & = 8 \
y + z & = 6 \
z & = - 6
\end{aligned} \right.$  (56, 12, -6) 
B)   $\left\{ \begin{aligned}
x - y + 6 z & = 8 \
y - z & = - 6 \
z & = 6
\end{aligned} \right.$  (44, 0, -6) 
C)   $\left\{ \begin{array} { r } 
x - y + 6 z = 8 \
y + z = 6 \
z = 6
\end{array} \right.$  (-40, 12, 6) 
D)   $\left\{ \begin{aligned}
x - y + 6 z & = 8 \
y - z & = 6 \
z & = - 6
\end{aligned} \right.$  (44, 0, -6) 
E)   $\left\{ \begin{aligned}
x - y + 6 z & = 8 \
y + z & = - 6 \
z & = 6
\end{aligned} \right.$  (-40, 12, 6) 
A)   $\left\{ \begin{aligned}
x - y + 6 z & = 8 \
y + z & = 6 \
z & = - 6
\end{aligned} \right.$  (56, 12, -6) 
B)   $\left\{ \begin{aligned}
x - y + 6 z & = 8 \
y - z & = - 6 \
z & = 6
\end{aligned} \right.$  (44, 0, -6) 
C)   $\left\{ \begin{array} { r } 
x - y + 6 z = 8 \
y + z = 6 \
z = 6
\end{array} \right.$  (-40, 12, 6) 
D)   $\left\{ \begin{aligned}
x - y + 6 z & = 8 \
y - z & = 6 \
z & = - 6
\end{aligned} \right.$  (44, 0, -6) 
E)   $\left\{ \begin{aligned}
x - y + 6 z & = 8 \
y + z & = - 6 \
z & = 6
\end{aligned} \right.$  (-40, 12, 6)

Question 13

Find all the cofactors of the matrix.   $\left[ \begin{array} { c c } 
0 & 6 \
3 & - 6
\end{array} \right]$

Accepted Answer

A)  $C _ { 11 } = - 6 , C _ { 12 } = - 3 , C _ { 21 } = - 6 , C _ { 22 } = 0$ 
B)  $C _ { 11 } = 6 , C _ { 12 } = 3 , C _ { 21 } = 6 , C _ { 22 } = 0$ 
C)  $C _ { 11 } = - 6 , C _ { 12 } = - 3 , C _ { 21 } = 0 , C _ { 22 } = 6$ 
D)  $C _ { 11 } = 0 , C _ { 12 } = - 3 , C _ { 21 } = - 6 , C _ { 22 } = 6$ 
E)  $C _ { 11 } = 0 , C _ { 12 } = - 6 , C _ { 21 } = - 3 , C _ { 22 } = 6$ 
A)  $C _ { 11 } = - 6 , C _ { 12 } = - 3 , C _ { 21 } = - 6 , C _ { 22 } = 0$ 
B)  $C _ { 11 } = 6 , C _ { 12 } = 3 , C _ { 21 } = 6 , C _ { 22 } = 0$ 
C)  $C _ { 11 } = - 6 , C _ { 12 } = - 3 , C _ { 21 } = 0 , C _ { 22 } = 6$ 
D)  $C _ { 11 } = 0 , C _ { 12 } = - 3 , C _ { 21 } = - 6 , C _ { 22 } = 6$ 
E)  $C _ { 11 } = 0 , C _ { 12 } = - 6 , C _ { 21 } = - 3 , C _ { 22 } = 6$

Question 14

Use determinants to find the area of a triangle with the given vertices and confirm your answer by plotting the points in a coordinate plane and using the formula $$\text { Area } \left. = \frac { 1 } { 2 } \text { (base)(height } \right)$$ .  
(5, -3), (2, -2), (7, 5)

Accepted Answer

A) 21 
B) 13 
C) 15 
D) 19 
E) 17 
A) 21 
B) 13 
C) 15 
D) 19 
E) 17

Question 15

Solve for x given the following equation involving a determinant. $\left| \begin{array} { c c } 
x - 6 & - 8 \
- 1 & x - 4
\end{array} \right| = 0$

Accepted Answer

A)  $x = 2,8$ 
B)  $x = - 2,8$ 
C)  $x = 4,16$ 
D)  $x = 2 , - 8$ 
E)  $x = - 2 , - 8$ 
A)  $x = 2,8$ 
B)  $x = - 2,8$ 
C)  $x = 4,16$ 
D)  $x = 2 , - 8$ 
E)  $x = - 2 , - 8$

Question 16

Fill in the blank using elementary row operations to form a row-equivalent matrix. $$\left[ \begin{array} { r r r } 
- 1 & 5 & - 3 \
- 2 & 1 & 4
\end{array} \right]$$ $\left[\begin{array}{rrr}
-1 & 5 & -3 \
0 &\square & 10
\end{array}\right]$

Accepted Answer

A)  $\left[ \begin{array} { r r r } 
- 1 & 5 & - 3 \
0 & - 9 & 10
\end{array} \right]$ 
B)  $\left[ \begin{array} { r r r } 
- 1 & 5 & - 3 \
0 & 0 & 10
\end{array} \right]$ 
C)  $\left[ \begin{array} { r r r } 
- 1 & 5 & - 3 \
0 & 11 & 10
\end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } 
- 1 & 5 & - 3 \
0 & 5 & 10
\end{array} \right]$ 
E)  $\left[ \begin{array} { r r r } 
- 1 & 5 & - 3 \
0 & 9 & 10
\end{array} \right]$ 
A)  $\left[ \begin{array} { r r r } 
- 1 & 5 & - 3 \
0 & - 9 & 10
\end{array} \right]$ 
B)  $\left[ \begin{array} { r r r } 
- 1 & 5 & - 3 \
0 & 0 & 10
\end{array} \right]$ 
C)  $\left[ \begin{array} { r r r } 
- 1 & 5 & - 3 \
0 & 11 & 10
\end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } 
- 1 & 5 & - 3 \
0 & 5 & 10
\end{array} \right]$ 
E)  $\left[ \begin{array} { r r r } 
- 1 & 5 & - 3 \
0 & 9 & 10
\end{array} \right]$

Question 17

Find the determinant of the matrix by the method of expansion by cofactors.Expand using the row 1.   $\left[ \begin{array} { c c c } 
- 3 & 2 & 1 \
4 & 5 & 6 \
2 & - 3 & 1
\end{array} \right]$

Accepted Answer

A) -73 
B) 75 
C) -75 
D) -74 
E) -76 
A) -73 
B) 75 
C) -75 
D) -74 
E) -76

Question 18

Use Cramer's Rule to solve (if possible) the system of equations.   $$\left\{ \begin{array} { l } 
x + 2 y + 3 z = - 5 \
- 2 x + y - z = 10 \
3 x - 3 y + 2 z = - 21
\end{array} \right.$$

Accepted Answer

A) (-2, 3, -3) 
B) (7, -3, 2) 
C) (2, 3, -3) 
D) (2, -3, -3) 
E) (-2, -3, -3) 
A) (-2, 3, -3) 
B) (7, -3, 2) 
C) (2, 3, -3) 
D) (2, -3, -3) 
E) (-2, -3, -3)

Question 19

Find 3A - 2B.   $$A = \left[ \begin{array} { c c c } 
- 3 & 5 & 0 \
3 & - 4 & 2 \
5 & 6 & - 4 \
0 & 8 & - 6 \
- 2 & - 1 & 0
\end{array} \right] , B = \left[ \begin{array} { c c c } 
- 2 & 5 & 4 \
4 & - 2 & - 5 \
10 & - 8 & - 2 \
5 & 4 & - 4 \
0 & 2 & - 4
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c c c } 
- 5 & 5 & - 8 \
1 & - 8 & 16 \
- 5 & 34 & - 8 \
- 10 & 16 & - 10 \
- 6 & - 7 & 8
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
- 7 & 5 & - 8 \
- 10 & - 8 & - 10 \
- 5 & 34 & - 8 \
1 & 16 & 16 \
- 6 & - 7 & 8
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
- 1 & 5 & - 8 \
16 & - 8 & 1 \
- 5 & 34 & - 8 \
- 10 & 16 & - 10 \
- 6 & - 7 & 8
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 9 & 5 & - 8 \
1 & - 8 & 16 \
- 6 & 34 & 8 \
- 10 & 16 & - 10 \
- 5 & - 7 & - 8
\end{array} \right]$ 
E)   $\left[ \begin{array} { c c c } 
- 3 & 5 & - 8 \
1 & - 8 & 16 \
- 8 & 34 & - 5 \
- 10 & 16 & - 10 \
8 & - 7 & - 6
\end{array} \right]$ 
A)  $\left[ \begin{array} { c c c } 
- 5 & 5 & - 8 \
1 & - 8 & 16 \
- 5 & 34 & - 8 \
- 10 & 16 & - 10 \
- 6 & - 7 & 8
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
- 7 & 5 & - 8 \
- 10 & - 8 & - 10 \
- 5 & 34 & - 8 \
1 & 16 & 16 \
- 6 & - 7 & 8
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
- 1 & 5 & - 8 \
16 & - 8 & 1 \
- 5 & 34 & - 8 \
- 10 & 16 & - 10 \
- 6 & - 7 & 8
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 9 & 5 & - 8 \
1 & - 8 & 16 \
- 6 & 34 & 8 \
- 10 & 16 & - 10 \
- 5 & - 7 & - 8
\end{array} \right]$ 
E)   $\left[ \begin{array} { c c c } 
- 3 & 5 & - 8 \
1 & - 8 & 16 \
- 8 & 34 & - 5 \
- 10 & 16 & - 10 \
8 & - 7 & - 6
\end{array} \right]$

Question 20

An augmented matrix that represents a system of linear equations (in variables x, y, z and w if applicable) has been reduced using Gauss-Jordan elimination.Find the solution represented by the augmented matrix.   $\left[ \begin{array} { c c c c } 
1&0 &\vdots& 5 \
0&1&\vdots& -6 \
\end{array} \right]$

Accepted Answer

A) (-6, 7) 
B) (-6, 5) 
C) (-5, 6) 
D) (5, -6) 
E) (6, 5) 
A) (-6, 7) 
B) (-6, 5) 
C) (-5, 6) 
D) (5, -6) 
E) (6, 5)

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

Fill in the blank using elementary row operations to form a row-equivalent matrix. $\left[ \begin{array} { r r r } 2 & 1 & - 2 \\- 10 & - 1 & 9\end{array} \right]$ $\left[ \begin{array} { l l l } 2 & 1 & - 2 \\0 & \square & - 1\end{array} \right]$

Find A + B. $A = \left[ \begin{array} { l l l } 5 & 5 & 4 \\9 & 9 & 2\end{array} \right] , B = \left[ \begin{array} { l l l } 3 & 7 & 8 \\7 & 6 & 4\end{array} \right]$

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

Write the matrix in reduced row-echelon form. $\left[ \begin{array} { r r r r } 9 & - 6 & - 3 & - 33 \\8 & - 1 & 1 & - 47 \\- 7 & - 7 & 6 & - 8\end{array} \right]$

Use a determinant to find y such that $( 6 , - 15 ) , ( 12 , y )$ , and $( 15 , - 6 )$ are collinear.

Find the product. $\left[ \begin{array} { l l } 9 & 3 \\7 & 8\end{array} \right] \left[ \begin{array} { l l } 8 & 7 \\8 & 4\end{array} \right]$

Find the determinant of the matrix by the method of expansion by cofactors.Expand using the column 2. $\left[ \begin{array} { c c c } - 3 & 2 & 1 \\4 & 5 & 6 \\2 & - 3 & 1\end{array} \right]$

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { r } 5 x - 4 y + z = - 12 \\- x + 2 y - 2 z = 12 \\3 x + y + z = - 2\end{array} \right.$

Use a determinant to determine whether the points (-3, -9), (-5, -11) and (0, -7) are collinear.

Find the system of linear equations represented by the augmented matrix.Then use back substitution to solve.(Use variables x, y, z and if applicable.) $\left[ \begin{array} { c c c c } 1&-1&6 &\vdots& 8 \\0&1&-1 &\vdots& 6 \\0 & 0 & 1 &\vdots&-6\\\end{array} \right]$

Find all the cofactors of the matrix. $\left[ \begin{array} { c c } 0 & 6 \\3 & - 6\end{array} \right]$

Use determinants to find the area of a triangle with the given vertices and confirm your answer by plotting the points in a coordinate plane and using the formula $\text { Area } \left. = \frac { 1 } { 2 } \text { (base)(height } \right)$ . (5, -3), (2, -2), (7, 5)

Solve for x given the following equation involving a determinant. $\left| \begin{array} { c c } x - 6 & - 8 \\- 1 & x - 4\end{array} \right| = 0$

Fill in the blank using elementary row operations to form a row-equivalent matrix. $\left[ \begin{array} { r r r } - 1 & 5 & - 3 \\- 2 & 1 & 4\end{array} \right]$ $\left[\begin{array}{rrr}-1 & 5 & -3 \\0 &\square & 10\end{array}\right]$

Find the determinant of the matrix by the method of expansion by cofactors.Expand using the row 1. $\left[ \begin{array} { c c c } - 3 & 2 & 1 \\4 & 5 & 6 \\2 & - 3 & 1\end{array} \right]$

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { l } x + 2 y + 3 z = - 5 \\- 2 x + y - z = 10 \\3 x - 3 y + 2 z = - 21\end{array} \right.$

Find 3A - 2B. $A = \left[ \begin{array} { c c c } - 3 & 5 & 0 \\3 & - 4 & 2 \\5 & 6 & - 4 \\0 & 8 & - 6 \\- 2 & - 1 & 0\end{array} \right] , B = \left[ \begin{array} { c c c } - 2 & 5 & 4 \\4 & - 2 & - 5 \\10 & - 8 & - 2 \\5 & 4 & - 4 \\0 & 2 & - 4\end{array} \right]$

An augmented matrix that represents a system of linear equations (in variables x, y, z and w if applicable) has been reduced using Gauss-Jordan elimination.Find the solution represented by the augmented matrix. $\left[ \begin{array} { c c c c } 1&0 &\vdots& 5 \\0&1&\vdots& -6 \\\end{array} \right]$

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

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Limits and An Introduction To Calculus

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Exam 8: Matrices and Determinants

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

Find A + B. ​ A=[554992],B=[378764]A = \left[ \begin{array} { l l l } 5 & 5 & 4 \\9 & 9 & 2\end{array} \right] , B = \left[ \begin{array} { l l l } 3 & 7 & 8 \\7 & 6 & 4\end{array} \right]A=[59​59​42​],B=[37​76​84​]

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

Write the matrix in reduced row-echelon form. [9−6−3−338−11−47−7−76−8]\left[ \begin{array} { r r r r } 9 & - 6 & - 3 & - 33 \\8 & - 1 & 1 & - 47 \\- 7 & - 7 & 6 & - 8\end{array} \right]​98−7​−6−1−7​−316​−33−47−8​​

Use a determinant to find y such that (6,−15),(12,y)( 6 , - 15 ) , ( 12 , y )(6,−15),(12,y) , and (15,−6)( 15 , - 6 )(15,−6) are collinear.

Find the product. [9378][8784]\left[ \begin{array} { l l } 9 & 3 \\7 & 8\end{array} \right] \left[ \begin{array} { l l } 8 & 7 \\8 & 4\end{array} \right][97​38​][88​74​]

Find the determinant of the matrix by the method of expansion by cofactors.Expand using the column 2. [−3214562−31]\left[ \begin{array} { c c c } - 3 & 2 & 1 \\4 & 5 & 6 \\2 & - 3 & 1\end{array} \right]​−342​25−3​161​​

Use Cramer's Rule to solve (if possible) the system of equations. {5x−4y+z=−12−x+2y−2z=123x+y+z=−2\left\{ \begin{array} { r } 5 x - 4 y + z = - 12 \\- x + 2 y - 2 z = 12 \\3 x + y + z = - 2\end{array} \right.⎩⎨⎧​5x−4y+z=−12−x+2y−2z=123x+y+z=−2​

Use a determinant to determine whether the points (-3, -9), (-5, -11) and (0, -7) are collinear.

Find all the cofactors of the matrix. [063−6]\left[ \begin{array} { c c } 0 & 6 \\3 & - 6\end{array} \right][03​6−6​]

Solve for x given the following equation involving a determinant. ∣x−6−8−1x−4∣=0\left| \begin{array} { c c } x - 6 & - 8 \\- 1 & x - 4\end{array} \right| = 0​x−6−1​−8x−4​​=0

Find the determinant of the matrix by the method of expansion by cofactors.Expand using the row 1. [−3214562−31]\left[ \begin{array} { c c c } - 3 & 2 & 1 \\4 & 5 & 6 \\2 & - 3 & 1\end{array} \right]​−342​25−3​161​​

Use Cramer's Rule to solve (if possible) the system of equations. {x+2y+3z=−5−2x+y−z=103x−3y+2z=−21\left\{ \begin{array} { l } x + 2 y + 3 z = - 5 \\- 2 x + y - z = 10 \\3 x - 3 y + 2 z = - 21\end{array} \right.⎩⎨⎧​x+2y+3z=−5−2x+y−z=103x−3y+2z=−21​

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

Find A + B. $A = \left[ \begin{array} { l l l } 5 & 5 & 4 \\9 & 9 & 2\end{array} \right] , B = \left[ \begin{array} { l l l } 3 & 7 & 8 \\7 & 6 & 4\end{array} \right]$

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

Write the matrix in reduced row-echelon form. $\left[ \begin{array} { r r r r } 9 & - 6 & - 3 & - 33 \\8 & - 1 & 1 & - 47 \\- 7 & - 7 & 6 & - 8\end{array} \right]$

Use a determinant to find y such that $( 6 , - 15 ) , ( 12 , y )$ , and $( 15 , - 6 )$ are collinear.

Find the product. $\left[ \begin{array} { l l } 9 & 3 \\7 & 8\end{array} \right] \left[ \begin{array} { l l } 8 & 7 \\8 & 4\end{array} \right]$

Find the determinant of the matrix by the method of expansion by cofactors.Expand using the column 2. $\left[ \begin{array} { c c c } - 3 & 2 & 1 \\4 & 5 & 6 \\2 & - 3 & 1\end{array} \right]$

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { r } 5 x - 4 y + z = - 12 \\- x + 2 y - 2 z = 12 \\3 x + y + z = - 2\end{array} \right.$

Find all the cofactors of the matrix. $\left[ \begin{array} { c c } 0 & 6 \\3 & - 6\end{array} \right]$

Solve for x given the following equation involving a determinant. $\left| \begin{array} { c c } x - 6 & - 8 \\- 1 & x - 4\end{array} \right| = 0$

Find the determinant of the matrix by the method of expansion by cofactors.Expand using the row 1. $\left[ \begin{array} { c c c } - 3 & 2 & 1 \\4 & 5 & 6 \\2 & - 3 & 1\end{array} \right]$

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { l } x + 2 y + 3 z = - 5 \\- 2 x + y - z = 10 \\3 x - 3 y + 2 z = - 21\end{array} \right.$