Question 1

Find all (a) minors and (b) cofactors of the matrix $\left[ \begin{array} { c c c } 
- 3 & 2 & - 8 \
- 3 & - 2 & 6 \
1 & - 3 & - 6
\end{array} \right]$ . Show your work.

Accepted Answer

The answer of Find all (a) minors and (b) cofactors...

Question 2

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $\left[ \begin{array} { c c c } 
0.8 & - 2.4 & 2.4 \
3.2 & 1.6 & 0 \
- 1.6 & - 5.6 & 4
\end{array} \right]$ .

Accepted Answer

A)  $- 1.28$ 
B)  $- 2.56$ 
C)  $- 0.512$ 
D)  $1.536$ 
E)  $- 1.024$ 
A)  $- 1.28$ 
B)  $- 2.56$ 
C)  $- 0.512$ 
D)  $1.536$ 
E)  $- 1.024$

Question 3

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. $\left\{ \begin{aligned}
x - 7 y - 2 z & = 43 \
y + 2 z & = - 14 \
z & = - 4
\end{aligned} \right.$ $\left\{ \begin{aligned}
x + 2 y + 3 z & = - 27 \
y - 2 z & = 2 \
z & = - 4
\end{aligned} \right.$

Accepted Answer

A) x = -7, y = -6, z = -4 
B) x = 7, y = 6, z = -4 
C) x = 6, y = -4, z = -7 
D) x = -6, y = -4, z = -7 
E) The systems yield different solutions. 
A) x = -7, y = -6, z = -4 
B) x = 7, y = 6, z = -4 
C) x = 6, y = -4, z = -7 
D) x = -6, y = -4, z = -7 
E) The systems yield different solutions.

Question 4

Solve for X in the equation given. $3 X = - 3 A - B , A = \left[ \begin{array} { c c } 
- 9 & - 9 \
3 & - 6
\end{array} \right] \text { and } B = \left[ \begin{array} { c c } 
21 & 12 \
3 & 30
\end{array} \right]$

Accepted Answer

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

Question 5

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

Accepted Answer

A) x = 4, y = 9 
B) x = -6, y = 7 
C) x = -9, y = -4 
D) x = -4, y = -9 
E) x = -4, y = -4 
A) x = 4, y = 9 
B) x = -6, y = 7 
C) x = -9, y = -4 
D) x = -4, y = -9 
E) x = -4, y = -4

Question 6

The following cryptogram was encoded with a 2 × 2 matrix. $\begin{array} { c c c c c l c c c c } 
215 & 260 & 110 & - 85 & 45 & 5 & 220 & 80 & 190 & 285 \
195 & 280 & 245 & 305 & 60 & - 60 & 200 & 25 & 145 & - 20
\end{array}$ The last word of the message is "_LIVES". What is the message?

Accepted Answer

A) CATS HAVE NINE LIVES 
B) SHE SAID ELVIS LIVES 
C) ALL GAVE THEIR LIVES 
D) SEATBELTS SAVE LIVES 
E) NO LIES CHANGE LIVES 
A) CATS HAVE NINE LIVES 
B) SHE SAID ELVIS LIVES 
C) ALL GAVE THEIR LIVES 
D) SEATBELTS SAVE LIVES 
E) NO LIES CHANGE LIVES

Question 7

Fill in the blank using elementary row operations to form a row-equivalent matrix. $$\begin{array} { l } 
{ \left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
- 6 & - 2 & - 1
\end{array} \right] } \
{ \left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
0 & \square & 5
\end{array} \right] }
\end{array}$$

Accepted Answer

A)  $\left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
0 & - 17 & 5
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
0 & 0 & 5
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
0 & 13 & 5
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
0 & 5 & 5
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
0 & 17 & 5
\end{array} \right]$ 
A)  $\left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
0 & - 17 & 5
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
0 & 0 & 5
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
0 & 13 & 5
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
0 & 5 & 5
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c c } 
- 2 & 5 & - 2 \
0 & 17 & 5
\end{array} \right]$

Question 8

Evaluate the determinant by expanding by cofactors. $\left| \begin{array} { c c c } 
2 & - 5 & - 3 \
1 & - 1 & - 6 \
- 1 & 5 & - 6
\end{array} \right|$

Accepted Answer

A)  $60$ 
B)  $36$ 
C)  $0$ 
D)  $- 24$ 
E)  $- 120$ 
A)  $60$ 
B)  $36$ 
C)  $0$ 
D)  $- 24$ 
E)  $- 120$

Question 9

Find the inverse of the matrix below, if it exists. $\left[ \begin{array} { c c } 
- 8 & 1 \
- 1 & - 5
\end{array} \right]$

Accepted Answer

A)  $\frac { 1 } { 41 } \left[ \begin{array} { c c } 
- 5 & - 1 \
1 & - 8
\end{array} \right]$ 
B)  $\frac { 1 } { 39 } \left[ \begin{array} { c c } 
- 5 & - 1 \
1 & - 8
\end{array} \right]$ 
C)  $\frac { 1 } { 41 } \left[ \begin{array} { c c } 
- 8 & - 1 \
1 & - 5
\end{array} \right]$ 
D)  $\frac { 1 } { 39 } \left[ \begin{array} { c c } 
- 8 & - 1 \
1 & - 5
\end{array} \right]$ 
E) The inverse of the matrix does not exist. 
A)  $\frac { 1 } { 41 } \left[ \begin{array} { c c } 
- 5 & - 1 \
1 & - 8
\end{array} \right]$ 
B)  $\frac { 1 } { 39 } \left[ \begin{array} { c c } 
- 5 & - 1 \
1 & - 8
\end{array} \right]$ 
C)  $\frac { 1 } { 41 } \left[ \begin{array} { c c } 
- 8 & - 1 \
1 & - 5
\end{array} \right]$ 
D)  $\frac { 1 } { 39 } \left[ \begin{array} { c c } 
- 8 & - 1 \
1 & - 5
\end{array} \right]$ 
E) The inverse of the matrix does not exist.

Question 10

Find the determinant of the matrix $\left[ \begin{array} { l l } 
- \frac { 1 } { 2 } & \frac { 5 } { 6 } \\
- 3 & \frac { 2 } { 5 }
\end{array} \right]$ .

Accepted Answer

A)  $\frac { 7 } { 6 }$ 
B)  $- \frac { 27 } { 10 }$ 
C)  $- \frac { 7 } { 12 }$ 
D)  $\frac { 23 } { 10 }$ 
E)  $\frac { 1 } { 4 }$ 
A)  $\frac { 7 } { 6 }$ 
B)  $- \frac { 27 } { 10 }$ 
C)  $- \frac { 7 } { 12 }$ 
D)  $\frac { 23 } { 10 }$ 
E)  $\frac { 1 } { 4 }$

Question 11

Use a determinant to determine whether the points $( 5 , - 9 ) , ( 2 , - 11 ) \text { and } ( 11 , - 5 )$ are collinear. Show all work.

Accepted Answer

@#IMG-DLM& ; therefo

Question 12

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

Accepted Answer

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

Question 13

If possible, find A - B. $A = \left[ \begin{array} { c c } 
9 & 8 \
6 & - 6
\end{array} \right] , B = \left[ \begin{array} { c c } 
- 8 & - 3 \
2 & 8
\end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { c c } 
17 & 11 \
4 & - 14
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 
1 & 5 \
8 & 2
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
17 & 8 \
6 & - 14
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
- 88 & - 91 \
- 36 & 30
\end{array} \right]$ 
E) not possible 
A)  $\left[ \begin{array} { c c } 
17 & 11 \
4 & - 14
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 
1 & 5 \
8 & 2
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
17 & 8 \
6 & - 14
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
- 88 & - 91 \
- 36 & 30
\end{array} \right]$ 
E) not possible

Question 14

Use the matrix capabilities of a graphing utility to find AB, if possible. $A = \left[ \begin{array} { c c } 
3 & 2 \
6 & 2 \
1 & - 4 \
5 & 7
\end{array} \right] , B = \left[ \begin{array} { c c c c } 
- 8 & 4 & - 1 & - 2 \
3 & 3 & - 4 & - 9
\end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { c c c c } 
- 18 & 18 & - 11 & - 24 \
- 42 & 30 & - 14 & - 30 \
- 20 & - 8 & 15 & 34 \
- 19 & 41 & - 33 & - 73
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c c } 
- 18 & - 42 & - 20 & - 19 \
18 & 30 & - 8 & 41 \
- 11 & - 14 & 15 & - 33 \
- 24 & - 30 & 34 & - 73
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c c } 
- 30 & 6 & 5 & 12 \
- 54 & 18 & 2 & 6 \
4 & 16 & - 17 & - 38 \
- 61 & - 1 & 23 & 53
\end{array} \right]$ 
D)  $\left[ \begin{array} { c } 
- 18 \
30 \
- 29 \
- 73
\end{array} \right]$ 
E) not possible 
A)  $\left[ \begin{array} { c c c c } 
- 18 & 18 & - 11 & - 24 \
- 42 & 30 & - 14 & - 30 \
- 20 & - 8 & 15 & 34 \
- 19 & 41 & - 33 & - 73
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c c } 
- 18 & - 42 & - 20 & - 19 \
18 & 30 & - 8 & 41 \
- 11 & - 14 & 15 & - 33 \
- 24 & - 30 & 34 & - 73
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c c } 
- 30 & 6 & 5 & 12 \
- 54 & 18 & 2 & 6 \
4 & 16 & - 17 & - 38 \
- 61 & - 1 & 23 & 53
\end{array} \right]$ 
D)  $\left[ \begin{array} { c } 
- 18 \
30 \
- 29 \
- 73
\end{array} \right]$ 
E) not possible

Question 15

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $$\left[ \begin{array} { c c c l l } 
- 2 & 2 & 0 & - 4 & 6 \
0 & 0 & 2 & 8 & 0 \
0 & - 4 & - 6 & - 10 & 2 \
2 & 0 & 0 & 14 & 0 \
0 & 0 & 0 & 2 & 0
\end{array} \right]$$ .

Accepted Answer

A)  $- 48$ 
B)  $224$ 
C)  $- 320$ 
D)  $- 384$ 
E)  $128$ 
A)  $- 48$ 
B)  $224$ 
C)  $- 320$ 
D)  $- 384$ 
E)  $128$

Question 16

Use the matrix capabilities of a graphing utility to solve the following system of linear equations: $\left\{ \begin{aligned}
- 16 x + 16 y + 24 z & = 2 \
8 x - 8 y & = - 1 \
8 y + 32 z & = 5
\end{aligned} \right.$

Accepted Answer

A)  $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 4 } \
0 \
- \frac { 3 } { 8 }
\end{array} \right]$ 
B)  $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 3 } { 8 } \\
- \frac { 1 } { 4 } \\
\frac { 1 } { 4 }
\end{array} \right]$ 
C)  $\left[ \begin{array} { c } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
- \frac { 3 } { 8 } \\
\frac { 1 } { 4 } \\
\frac { 1 } { 8 }
\end{array} \right]$ 
D)  $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 8 } \\
- \frac { 1 } { 2 } \\
\frac { 1 } { 4 }
\end{array} \right]$ 
E)  $\left[ \begin{array} { c } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 2 } \\
\frac { 5 } { 8 } \\
0
\end{array} \right]$ 
A)  $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 4 } \
0 \
- \frac { 3 } { 8 }
\end{array} \right]$ 
B)  $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 3 } { 8 } \\
- \frac { 1 } { 4 } \\
\frac { 1 } { 4 }
\end{array} \right]$ 
C)  $\left[ \begin{array} { c } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
- \frac { 3 } { 8 } \\
\frac { 1 } { 4 } \\
\frac { 1 } { 8 }
\end{array} \right]$ 
D)  $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 8 } \\
- \frac { 1 } { 2 } \\
\frac { 1 } { 4 }
\end{array} \right]$ 
E)  $\left[ \begin{array} { c } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 2 } \\
\frac { 5 } { 8 } \\
0
\end{array} \right]$

Question 17

Find the determinant of the matrix below. Expand by cofactors on the row or column that appears to make the computations easiest. $$\left[ \begin{array} { c c c c } 
- 1 & 7 & 5 & 3 \
3 & 3 & - 7 & - 4 \
4 & - 5 & - 7 & 4 \
2 & 2 & - 4 & - 2
\end{array} \right]$$

Accepted Answer

A) 766 
B) 990 
C) 94 
D) 234 
E) -146 
A) 766 
B) 990 
C) 94 
D) 234 
E) -146

Question 18

The determinant $\left| \begin{array} { l l l } 
x & y & 1 \
x _ { 2 } & y _ { 2 } & 1 \
x _ { 3 } & y _ { 3 } & 1
\end{array} \right| = 0$ is the equation of a line through the points $\left( x _ { 1 } , y _ { 1 } \right)$ and $\left( x _ { 2 } , y _ { 2 } \right).$ Use this formula to find the equation of the line passing through the points $( - 1,8 )$ and $( 4 , - 7 )$

Accepted Answer

A)  $5 x - y = - 3$ 
B)  $3 x + y = - 5$ 
C)  $- 3 x + y = - 5$ 
D)  $3 x - y = - 5$ 
E)  $- 3 x - y = - 5$ 
A)  $5 x - y = - 3$ 
B)  $3 x + y = - 5$ 
C)  $- 3 x + y = - 5$ 
D)  $3 x - y = - 5$ 
E)  $- 3 x - y = - 5$

Question 19

Solve for X in the equation given. $16 A - 16 B = - 4 X , A = \left[ \begin{array} { c c c } 
5 & - 6 & 4 \
6 & 3 & - 5
\end{array} \right] \text { and } B = \left[ \begin{array} { c c c } 
- 2 & - 9 & - 1 \
- 2 & 4 & 3
\end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { c c c } 
112 & 48 & 80 \
128 & - 16 & - 128
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
3 & - 15 & 3 \
4 & 7 & - 2
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
- 28 & - 12 & - 20 \
- 32 & 4 & 32
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 12 & 60 & - 12 \
- 16 & - 28 & 8
\end{array} \right]$ 
E) not possible 
A)  $\left[ \begin{array} { c c c } 
112 & 48 & 80 \
128 & - 16 & - 128
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
3 & - 15 & 3 \
4 & 7 & - 2
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
- 28 & - 12 & - 20 \
- 32 & 4 & 32
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 12 & 60 & - 12 \
- 16 & - 28 & 8
\end{array} \right]$ 
E) not possible

Question 20

Find the determinant of $\left[ \begin{array} { c c c } 
0 & 4 & 4 \
8 & - 8 & 0 \
- 4 & - 32 & - 12
\end{array} \right]$ .

Accepted Answer

A)  $- 736$ 
B)  $- 828$ 
C)  $- 768$ 
D)  $512$ 
E)  $- 64$ 
A)  $- 736$ 
B)  $- 828$ 
C)  $- 768$ 
D)  $512$ 
E)  $- 64$

Find all (a) minors and (b) cofactors of the matrix $\left[ \begin{array} { c c c } - 3 & 2 & - 8 \\- 3 & - 2 & 6 \\1 & - 3 & - 6\end{array} \right]$ . Show your work.

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $\left[ \begin{array} { c c c } 0.8 & - 2.4 & 2.4 \\3.2 & 1.6 & 0 \\- 1.6 & - 5.6 & 4\end{array} \right]$ .

Solve for X in the equation given. $3 X = - 3 A - B , A = \left[ \begin{array} { c c } - 9 & - 9 \\3 & - 6\end{array} \right] \text { and } B = \left[ \begin{array} { c c } 21 & 12 \\3 & 30\end{array} \right]$

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

The following cryptogram was encoded with a 2 × 2 matrix. 215 260 110 -85 45 5 220 80 190 285 195 280 245 305 60 -60 200 25 145 -20 The last word of the message is "_LIVES". What is the message?

Fill in the blank using elementary row operations to form a row-equivalent matrix. -2 5 -2 -6 -2 -1 -2 5 -2 0 \square 5

Evaluate the determinant by expanding by cofactors. $\left| \begin{array} { c c c } 2 & - 5 & - 3 \\1 & - 1 & - 6 \\- 1 & 5 & - 6\end{array} \right|$

Find the inverse of the matrix below, if it exists. $\left[ \begin{array} { c c } - 8 & 1 \\- 1 & - 5\end{array} \right]$

Find the determinant of the matrix $\left[ \begin{array} { l l } - \frac { 1 } { 2 } & \frac { 5 } { 6 } \\\\- 3 & \frac { 2 } { 5 }\end{array} \right]$ .

Use a determinant to determine whether the points $( 5 , - 9 ) , ( 2 , - 11 ) \text { and } ( 11 , - 5 )$ are collinear. Show all work.

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

If possible, find A - B. $A = \left[ \begin{array} { c c } 9 & 8 \\6 & - 6\end{array} \right] , B = \left[ \begin{array} { c c } - 8 & - 3 \\2 & 8\end{array} \right]$

Use the matrix capabilities of a graphing utility to find AB, if possible. $A = \left[ \begin{array} { c c } 3 & 2 \\6 & 2 \\1 & - 4 \\5 & 7\end{array} \right] , B = \left[ \begin{array} { c c c c } - 8 & 4 & - 1 & - 2 \\3 & 3 & - 4 & - 9\end{array} \right]$

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $\left[ \begin{array} { c c c l l } - 2 & 2 & 0 & - 4 & 6 \\0 & 0 & 2 & 8 & 0 \\0 & - 4 & - 6 & - 10 & 2 \\2 & 0 & 0 & 14 & 0 \\0 & 0 & 0 & 2 & 0\end{array} \right]$ .

Use the matrix capabilities of a graphing utility to solve the following system of linear equations: $\left\{ \begin{aligned}- 16 x + 16 y + 24 z & = 2 \\8 x - 8 y & = - 1 \\8 y + 32 z & = 5\end{aligned} \right.$

Find the determinant of the matrix below. Expand by cofactors on the row or column that appears to make the computations easiest. $\left[ \begin{array} { c c c c } - 1 & 7 & 5 & 3 \\3 & 3 & - 7 & - 4 \\4 & - 5 & - 7 & 4 \\2 & 2 & - 4 & - 2\end{array} \right]$

Solve for X in the equation given. $16 A - 16 B = - 4 X , A = \left[ \begin{array} { c c c } 5 & - 6 & 4 \\6 & 3 & - 5\end{array} \right] \text { and } B = \left[ \begin{array} { c c c } - 2 & - 9 & - 1 \\- 2 & 4 & 3\end{array} \right]$

Find the determinant of $\left[ \begin{array} { c c c } 0 & 4 & 4 \\8 & - 8 & 0 \\- 4 & - 32 & - 12\end{array} \right]$ .

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

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Exam 7: Matrices and Determinants

Find all (a) minors and (b) cofactors of the matrix [−32−8−3−261−3−6]\left[ \begin{array} { c c c } - 3 & 2 & - 8 \\- 3 & - 2 & 6 \\1 & - 3 & - 6\end{array} \right]​−3−31​2−2−3​−86−6​​ . Show your work.

Use the matrix capabilities of a graphing utility to find the determinant of the matrix [0.8−2.42.43.21.60−1.6−5.64]\left[ \begin{array} { c c c } 0.8 & - 2.4 & 2.4 \\3.2 & 1.6 & 0 \\- 1.6 & - 5.6 & 4\end{array} \right]​0.83.2−1.6​−2.41.6−5.6​2.404​​ .

Find x and y. [−6xy7]=[−6−4−97]\left[ \begin{array} { c c } - 6 & x \\y & 7\end{array} \right] = \left[ \begin{array} { c c } - 6 & - 4 \\- 9 & 7\end{array} \right][−6y​x7​]=[−6−9​−47​]

The following cryptogram was encoded with a 2 × 2 matrix. 215 260 110 -85 45 5 220 80 190 285 195 280 245 305 60 -60 200 25 145 -20 The last word of the message is "_LIVES". What is the message?

Fill in the blank using elementary row operations to form a row-equivalent matrix. -2 5 -2 -6 -2 -1 -2 5 -2 0 \square 5

Evaluate the determinant by expanding by cofactors. ∣2−5−31−1−6−15−6∣\left| \begin{array} { c c c } 2 & - 5 & - 3 \\1 & - 1 & - 6 \\- 1 & 5 & - 6\end{array} \right|​21−1​−5−15​−3−6−6​​

Find the inverse of the matrix below, if it exists. [−81−1−5]\left[ \begin{array} { c c } - 8 & 1 \\- 1 & - 5\end{array} \right][−8−1​1−5​]

Find the determinant of the matrix [−1256−325]\left[ \begin{array} { l l } - \frac { 1 } { 2 } & \frac { 5 } { 6 } \\\\- 3 & \frac { 2 } { 5 }\end{array} \right]​−21​−3​65​52​​​ .

Use a determinant to determine whether the points (5,−9),(2,−11) and (11,−5)( 5 , - 9 ) , ( 2 , - 11 ) \text { and } ( 11 , - 5 )(5,−9),(2,−11) and (11,−5) are collinear. Show all work.

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​] .

If possible, find A - B. A=[986−6],B=[−8−328]A = \left[ \begin{array} { c c } 9 & 8 \\6 & - 6\end{array} \right] , B = \left[ \begin{array} { c c } - 8 & - 3 \\2 & 8\end{array} \right]A=[96​8−6​],B=[−82​−38​]

Find the determinant of [0448−80−4−32−12]\left[ \begin{array} { c c c } 0 & 4 & 4 \\8 & - 8 & 0 \\- 4 & - 32 & - 12\end{array} \right]​08−4​4−8−32​40−12​​ .

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

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Find all (a) minors and (b) cofactors of the matrix $\left[ \begin{array} { c c c } - 3 & 2 & - 8 \\- 3 & - 2 & 6 \\1 & - 3 & - 6\end{array} \right]$ . Show your work.

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $\left[ \begin{array} { c c c } 0.8 & - 2.4 & 2.4 \\3.2 & 1.6 & 0 \\- 1.6 & - 5.6 & 4\end{array} \right]$ .

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

Evaluate the determinant by expanding by cofactors. $\left| \begin{array} { c c c } 2 & - 5 & - 3 \\1 & - 1 & - 6 \\- 1 & 5 & - 6\end{array} \right|$

Find the inverse of the matrix below, if it exists. $\left[ \begin{array} { c c } - 8 & 1 \\- 1 & - 5\end{array} \right]$

Find the determinant of the matrix $\left[ \begin{array} { l l } - \frac { 1 } { 2 } & \frac { 5 } { 6 } \\\\- 3 & \frac { 2 } { 5 }\end{array} \right]$ .

Use a determinant to determine whether the points $( 5 , - 9 ) , ( 2 , - 11 ) \text { and } ( 11 , - 5 )$ are collinear. Show all work.

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

If possible, find A - B. $A = \left[ \begin{array} { c c } 9 & 8 \\6 & - 6\end{array} \right] , B = \left[ \begin{array} { c c } - 8 & - 3 \\2 & 8\end{array} \right]$

Find the determinant of $\left[ \begin{array} { c c c } 0 & 4 & 4 \\8 & - 8 & 0 \\- 4 & - 32 & - 12\end{array} \right]$ .