Question 1

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 2

Find y such that the points are collinear. ​
(-3, 3), (-4, y), (-2, 4)
​

Accepted Answer

A) y = 3 
B) y = 4 
C) y = -2 
D) y = 2 
E) y = 5 
A) y = 3 
B) y = 4 
C) y = -2 
D) y = 2 
E) y = 5

Question 3

Find all minors of the matrix.   $\left[ \begin{array} { c c } 
4 & 5 \
4 & - 8
\end{array} \right]$

Accepted Answer

A)  $M _ { 11 } = - 8 , M _ { 12 } = - 4 , M _ { 21 } = - 5 , M _ { 22 } = - 4$ 
B)  $M _ { 11 } = 5 , M _ { 12 } = 4 , M _ { 21 } = 8 , M _ { 22 } = 4$ 
C)  $M _ { 11 } = 4 , M _ { 12 } = 8 , M _ { 21 } = 4 , M _ { 22 } = 5$ 
D)  $M _ { 11 } = - 8 , M _ { 12 } = 4 , M _ { 21 } = 5 , M _ { 22 } = 4$ 
E)  $M _ { 11 } = 4 , M _ { 12 } = 5 , M _ { 21 } = 4 , M _ { 22 } = 8$ 
A)  $M _ { 11 } = - 8 , M _ { 12 } = - 4 , M _ { 21 } = - 5 , M _ { 22 } = - 4$ 
B)  $M _ { 11 } = 5 , M _ { 12 } = 4 , M _ { 21 } = 8 , M _ { 22 } = 4$ 
C)  $M _ { 11 } = 4 , M _ { 12 } = 8 , M _ { 21 } = 4 , M _ { 22 } = 5$ 
D)  $M _ { 11 } = - 8 , M _ { 12 } = 4 , M _ { 21 } = 5 , M _ { 22 } = 4$ 
E)  $M _ { 11 } = 4 , M _ { 12 } = 5 , M _ { 21 } = 4 , M _ { 22 } = 8$

Question 4

Given $A = \left[ \begin{array} { c c c } 
0 & 2 & 4 \
- 6 & - 4 & 2 \
0 & 8 & 2
\end{array} \right]$ , find $| A |$ .

Accepted Answer

A) -168 
B) -80 
C) -88 
D) 40 
E) 120 
A) -168 
B) -80 
C) -88 
D) 40 
E) 120

Question 5

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 6

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

Accepted Answer

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

Question 7

Solve the system using Gauss-Jordan elimination.   $$\left\{ \begin{array} { l } 
w + x = 9 \
w + y = 0 \
x + z = 0
\end{array} \right.$$

Accepted Answer

A)  $w = - 9 + z , x = - z , y = - 9 - z , z$ 
B)  $w = 9 + z , x = - z , y = - 9 - z , z$ 
C)  $w = 9 + z , x = - z , y = 9 - z , z$ 
D)  $w = 9 + z , x = - z , y = 9 + z , z$ 
E)  $w = 9 - z , x = - z , y = - 9 - z , z$ 
A)  $w = - 9 + z , x = - z , y = - 9 - z , z$ 
B)  $w = 9 + z , x = - z , y = - 9 - z , z$ 
C)  $w = 9 + z , x = - z , y = 9 - z , z$ 
D)  $w = 9 + z , x = - z , y = 9 + z , z$ 
E)  $w = 9 - z , x = - z , y = - 9 - z , z$

Question 8

Determine whether the two systems of linear equations yield the same solutions.If so, find the solutions using matrices. $$\left\{ \begin{aligned}
x + 2 y - 4 z & = - 8 \
y - 7 z & = 4 \
z & = - 1
\end{aligned} \right.$$ $$\left\{ \begin{aligned}
x - y - 5 z & = 21 \
y - 6 z & = 3 \
z & = - 1
\end{aligned} \right.$$

Accepted Answer

A) x = 6, y = 3, z = -1 
B) x = -6, y = -3, z = -1 
C) x = 3, y = -1, z = -6 
D) The systems yield different solutions. 
E) x = -3, y = -1, z = -6 
A) x = 6, y = 3, z = -1 
B) x = -6, y = -3, z = -1 
C) x = 3, y = -1, z = -6 
D) The systems yield different solutions. 
E) x = -3, y = -1, z = -6

Question 9

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

Accepted Answer

A) -19 
B) -21 
C) -22 
D) -20 
E) -18 
A) -19 
B) -21 
C) -22 
D) -20 
E) -18

Question 10

Use determinants to find the area of the triangle with vertices at the given points. ​
P(0, 0), Q(4, 0), R(4, 3)
​

Accepted Answer

A) A = 12 
B) A = 4 
C) A = 6 
D) A = 3 
E) none of these 
A) A = 12 
B) A = 4 
C) A = 6 
D) A = 3 
E) none of these

Question 11

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.)   $$\left[ \begin{array} { c c c c } 
1 & -4 & \vdots & 6 \
0 & 1 & \vdots & - 5
\end{array} \right]$$

Accepted Answer

A)   $\left\{ \begin{aligned}
x + 4 y & = 6 \
y & = - 5
\end{aligned} \right.$ (-14, -5) 
B)   $\left\{ \begin{aligned}
x - 4 y & = 6 \
y & = - 5
\end{aligned} \right.$ (-14, -5) 
C)   $\left\{ \begin{aligned}
x - 4 y & = 6 \
y & = - 5
\end{aligned} \right.$  (-14, 5) 
D)  $\left\{ \begin{aligned}
x + 4 y & = 6 \
y & = - 5
\end{aligned} \right.$  (-5, -14) 
E)   $\left\{ \begin{aligned}
x - 4 y & = 6 \
y & = - 5
\end{aligned} \right.$  (-5, -14) 
A)   $\left\{ \begin{aligned}
x + 4 y & = 6 \
y & = - 5
\end{aligned} \right.$ (-14, -5) 
B)   $\left\{ \begin{aligned}
x - 4 y & = 6 \
y & = - 5
\end{aligned} \right.$ (-14, -5) 
C)   $\left\{ \begin{aligned}
x - 4 y & = 6 \
y & = - 5
\end{aligned} \right.$  (-14, 5) 
D)  $\left\{ \begin{aligned}
x + 4 y & = 6 \
y & = - 5
\end{aligned} \right.$  (-5, -14) 
E)   $\left\{ \begin{aligned}
x - 4 y & = 6 \
y & = - 5
\end{aligned} \right.$  (-5, -14)

Question 12

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.   $\left[ \begin{array} { c c c c } 
1&0&0&\vdots& -3 \
0&1&0&\vdots& -9 \
0 & 0 & 1 &\vdots&3\
\end{array} \right]$

Accepted Answer

A)  (-3, 9, 3) 
B) (3, -9, 3) 
C) (-3, -9, 3) 
D) (-3, -9, -3) 
E) (-3, 3, -9) 
A)  (-3, 9, 3) 
B) (3, -9, 3) 
C) (-3, -9, 3) 
D) (-3, -9, -3) 
E) (-3, 3, -9)

Question 13

Find x and y. $\left[ \begin{array} { c c } 
6 & x \
y & - 8
\end{array} \right] = \left[ \begin{array} { c c } 
6 & - 1 \
- 5 & - 8
\end{array} \right]$

Accepted Answer

A) x = -1, y = -5 
B) x = -1, y = -1 
C) x = 1, y = 5 
D) x = -5, y = -1 
E) x = 6, y = -8 
A) x = -1, y = -5 
B) x = -1, y = -1 
C) x = 1, y = 5 
D) x = -5, y = -1 
E) x = 6, y = -8

Question 14

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 15

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 16

Find the inverse of the matrix $\left[ \begin{array} { c c } 
2 & 4 \
- 6 & - 10
\end{array} \right]$ .

Accepted Answer

A)   $\frac { 1 } { 13 } \left[ \begin{array} { l l } 
- 2 & - 4 \
- 6 & 10
\end{array} \right]$ 
B)   $\left[ \begin{array} { c c } 
- 2 & - 4 \
10 & 6
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c } 
2 & - 6 \
8 & - 14
\end{array} \right]$ 
D)   $\frac { 1 } { 7 } \left[ \begin{array} { c c } 
2 & 4 \
- 6 & - 10
\end{array} \right]$ 
E)   $\frac { 1 } { 2 } \left[ \begin{array} { c c } 
- 5 & - 2 \
3 & 1
\end{array} \right]$ 
A)   $\frac { 1 } { 13 } \left[ \begin{array} { l l } 
- 2 & - 4 \
- 6 & 10
\end{array} \right]$ 
B)   $\left[ \begin{array} { c c } 
- 2 & - 4 \
10 & 6
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c } 
2 & - 6 \
8 & - 14
\end{array} \right]$ 
D)   $\frac { 1 } { 7 } \left[ \begin{array} { c c } 
2 & 4 \
- 6 & - 10
\end{array} \right]$ 
E)   $\frac { 1 } { 2 } \left[ \begin{array} { c c } 
- 5 & - 2 \
3 & 1
\end{array} \right]$

Question 17

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$\left\{ \begin{aligned}
4 x - 9 y - 9 z & = - 74 \
6 x + 8 y - 2 z & = - 22 \
6 x - y + 9 z & = 4
\end{aligned} \right.$$

Accepted Answer

A) x = 5, y = -2, z = 4 
B) x = 2, y = 5, z = -4 
C) x = -5, y = -2, z = 4 
D) no solution 
E) x = -5, y = 2, z = 4 
A) x = 5, y = -2, z = 4 
B) x = 2, y = 5, z = -4 
C) x = -5, y = -2, z = 4 
D) no solution 
E) x = -5, y = 2, z = 4

Question 18

Use a graphing utility and Cramer's Rule to solve (if possible) the system of equations.   $$\left\{ \begin{array} { r } 
x + 2 y - z = - 5 \
2 x - 2 y - 2 z = - 4 \
- x + 3 y + 4 z = 6
\end{array} \right.$$

Accepted Answer

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

Question 19

Use the matrix capabilities of a graphing utility to solve the following system of linear equations: $$\left\{ \begin{array} { c c c } 
15 x - 5 y & = & 3 \
10 x + 10 y & = & 6 \
20 z & = & - 12
\end{array} \right.$$

Accepted Answer

A)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 3 } { 10 } \
\frac { 9 } { 10 } \
\frac { - 3 } { 5 }
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 10 } \
0 \
\frac { - 3 } { 5 }
\end{array} \right]$ 
C)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 3 } { 10 } \
\frac { 3 } { 10 } \
\frac { - 3 } { 5 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 3 } { 10 } \
\frac { - 3 } { 10 } \
0
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 3 } { 10 } \
\frac { 3 } { 10 } \
\frac { - 3 } { 5 }
\end{array} \right]$ 
A)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 3 } { 10 } \
\frac { 9 } { 10 } \
\frac { - 3 } { 5 }
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 10 } \
0 \
\frac { - 3 } { 5 }
\end{array} \right]$ 
C)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 3 } { 10 } \
\frac { 3 } { 10 } \
\frac { - 3 } { 5 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 3 } { 10 } \
\frac { - 3 } { 10 } \
0
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 3 } { 10 } \
\frac { 3 } { 10 } \
\frac { - 3 } { 5 }
\end{array} \right]$

Question 20

Evaluate the determinant in which the entries are functions. 
 $\left| \begin{array} { c c } 
2 u & - 4 \
- 4 & 3 v
\end{array} \right|$

Accepted Answer

A)  $6 + 16 u v$ 
B)  $6 u v + 16$ 
C)  $6 u v - 16$ 
D)  $6 - 16 u v$ 
E)  $6 - u v$ 
A)  $6 + 16 u v$ 
B)  $6 u v + 16$ 
C)  $6 u v - 16$ 
D)  $6 - 16 u v$ 
E)  $6 - u v$

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

Find y such that the points are collinear. (-3, 3), (-4, y), (-2, 4)

Find all minors of the matrix. $\left[ \begin{array} { c c } 4 & 5 \\4 & - 8\end{array} \right]$

Given $A = \left[ \begin{array} { c c c } 0 & 2 & 4 \\- 6 & - 4 & 2 \\0 & 8 & 2\end{array} \right]$ , find $| A |$ .

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

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

Solve the system using Gauss-Jordan elimination. $\left\{ \begin{array} { l } w + x = 9 \\w + y = 0 \\x + z = 0\end{array} \right.$

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

Use determinants to find the area of the triangle with vertices at the given points. P(0, 0), Q(4, 0), R(4, 3)

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.) $\left[ \begin{array} { c c c c } 1 & -4 & \vdots & 6 \\0 & 1 & \vdots & - 5\end{array} \right]$

Find x and y. $\left[ \begin{array} { c c } 6 & x \\y & - 8\end{array} \right] = \left[ \begin{array} { c c } 6 & - 1 \\- 5 & - 8\end{array} \right]$

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

Find the inverse of the matrix $\left[ \begin{array} { c c } 2 & 4 \\- 6 & - 10\end{array} \right]$ .

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{aligned}4 x - 9 y - 9 z & = - 74 \\6 x + 8 y - 2 z & = - 22 \\6 x - y + 9 z & = 4\end{aligned} \right.$

Use a graphing utility and Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { r } x + 2 y - z = - 5 \\2 x - 2 y - 2 z = - 4 \\- x + 3 y + 4 z = 6\end{array} \right.$

Use the matrix capabilities of a graphing utility to solve the following system of linear equations: $\left\{ \begin{array} { c c c } 15 x - 5 y & = & 3 \\10 x + 10 y & = & 6 \\20 z & = & - 12\end{array} \right.$

Evaluate the determinant in which the entries are functions. $\left| \begin{array} { c c } 2 u & - 4 \\- 4 & 3 v\end{array} \right|$

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Exam 8: Matrices and Determinants

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 y such that the points are collinear. ​ (-3, 3), (-4, y), (-2, 4) ​

Find all minors of the matrix. [454−8]\left[ \begin{array} { c c } 4 & 5 \\4 & - 8\end{array} \right][44​5−8​]

Given A=[024−6−42082]A = \left[ \begin{array} { c c c } 0 & 2 & 4 \\- 6 & - 4 & 2 \\0 & 8 & 2\end{array} \right]A=​0−60​2−48​422​​ , find ∣A∣| A |∣A∣ .

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​​

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

Solve the system using Gauss-Jordan elimination. {w+x=9w+y=0x+z=0\left\{ \begin{array} { l } w + x = 9 \\w + y = 0 \\x + z = 0\end{array} \right.⎩⎨⎧​w+x=9w+y=0x+z=0​

Find ∣B∣| B |∣B∣ . B=[500−4]B = \left[ \begin{array} { c c } 5 & 0 \\0 & - 4\end{array} \right]B=[50​0−4​]

Use determinants to find the area of the triangle with vertices at the given points. ​ P(0, 0), Q(4, 0), R(4, 3) ​

Find x and y. [6xy−8]=[6−1−5−8]\left[ \begin{array} { c c } 6 & x \\y & - 8\end{array} \right] = \left[ \begin{array} { c c } 6 & - 1 \\- 5 & - 8\end{array} \right][6y​x−8​]=[6−5​−1−8​]

Find the inverse of the matrix [24−6−10]\left[ \begin{array} { c c } 2 & 4 \\- 6 & - 10\end{array} \right][2−6​4−10​] .

Use a graphing utility and Cramer's Rule to solve (if possible) the system of equations. {x+2y−z=−52x−2y−2z=−4−x+3y+4z=6\left\{ \begin{array} { r } x + 2 y - z = - 5 \\2 x - 2 y - 2 z = - 4 \\- x + 3 y + 4 z = 6\end{array} \right.⎩⎨⎧​x+2y−z=−52x−2y−2z=−4−x+3y+4z=6​

Use the matrix capabilities of a graphing utility to solve the following system of linear equations: {15x−5y=310x+10y=620z=−12\left\{ \begin{array} { c c c } 15 x - 5 y & = & 3 \\10 x + 10 y & = & 6 \\20 z & = & - 12\end{array} \right.⎩⎨⎧​15x−5y10x+10y20z​===​36−12​

Evaluate the determinant in which the entries are functions. ∣2u−4−43v∣\left| \begin{array} { c c } 2 u & - 4 \\- 4 & 3 v\end{array} \right|​2u−4​−43v​​