Question 1

Select the order for the following matrix.   $$\left[ \begin{array} { c c c c } 
- 3 & 5 & 10 & 0 \
0 & 0 & 5 & 5 \
4 & 4 & 8 & 5
\end{array} \right]$$

Accepted Answer

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

Question 2

Find y such that the points are collinear. ​
(5, -4), (3, y), (4, -3)
​

Accepted Answer

A) y = -2 
B) y = 1 
C) y = 0 
D) y = -1 
E) y = 2 
A) y = -2 
B) y = 1 
C) y = 0 
D) y = -1 
E) y = 2

Question 3

Find the inverse of A.
 $A = \left[ \begin{array} { c c c } 
- 4 & 12 & 4 \
8 & 20 & 0 \
12 & 4 & - 8
\end{array} \right]$

Accepted Answer

A)   $A ^ { - 1 } = \frac { 1 } { 36 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$ 
B)   $A ^ { - 1 } = \frac { 1 } { 24 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & 1 & 2 \
13 & 10 & 11
\end{array} \right]$ 
C)   $A ^ { - 1 } = \frac { 1 } { 28 } \left[ \begin{array} { c c c } 
- 10 & 7 & - 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$ 
D)   $A ^ { - 1 } = \frac { 1 } { 36 } \left[ \begin{array} { c c c } 
- 10 & 7 & - 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$ 
E)   $A ^ { - 1 } = \frac { 1 } { 28 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & 1 & 2 \
13 & 10 & 11
\end{array} \right]$ 
A)   $A ^ { - 1 } = \frac { 1 } { 36 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$ 
B)   $A ^ { - 1 } = \frac { 1 } { 24 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & 1 & 2 \
13 & 10 & 11
\end{array} \right]$ 
C)   $A ^ { - 1 } = \frac { 1 } { 28 } \left[ \begin{array} { c c c } 
- 10 & 7 & - 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$ 
D)   $A ^ { - 1 } = \frac { 1 } { 36 } \left[ \begin{array} { c c c } 
- 10 & 7 & - 5 \
4 & - 1 & 2 \
- 13 & 10 & - 11
\end{array} \right]$ 
E)   $A ^ { - 1 } = \frac { 1 } { 28 } \left[ \begin{array} { c c c } 
10 & 7 & 5 \
4 & 1 & 2 \
13 & 10 & 11
\end{array} \right]$

Question 4

​Find a value of y such that the triangle with the given vertices has an area of 26 square units. ​
​(-6, -6), ​(2, -2), ​(-9, y)
​

Accepted Answer

A) ​y = -1 or y = -14 
B) ​y = -1 or y = 14 
C) ​y = -1 or y = 0 
D) ​y = 1 or y = 14 
E) ​y = 1 or y = -14 
A) ​y = -1 or y = -14 
B) ​y = -1 or y = 14 
C) ​y = -1 or y = 0 
D) ​y = 1 or y = 14 
E) ​y = 1 or y = -14

Question 5

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.  
  $$\left\{ \begin{array} { r l r } 
3 x + 2 y - z + w & = 0 \
x - y + 4 z - w & = 28 \
- 2 x + y + 2 z - w & = 3 \
x + y + z + w & = 8
\end{array} \right.$$

Accepted Answer

A)  (-3, -2, 6, 1) 
B) (-3, 2, -6, -1) 
C) (3, -2, 6, 1) 
D) (3, -2, 6, -1) 
E) (3, -2, -6, 1) 
A)  (-3, -2, 6, 1) 
B) (-3, 2, -6, -1) 
C) (3, -2, 6, 1) 
D) (3, -2, 6, -1) 
E) (3, -2, -6, 1)

Question 6

Find the determinant of the matrix.  $\left[ \begin{array} { c c } 
7 & 1 \
- 1 & 2
\end{array} \right]$

Accepted Answer

A) 15 
B) 13 
C)  $- 13$ 
D)  $- \frac { 1 } { 13 }$ 
E)  $- 15$ 
A) 15 
B) 13 
C)  $- 13$ 
D)  $- \frac { 1 } { 13 }$ 
E)  $- 15$

Question 7

Use Cramer's Rule to solve (if possible) the system of equations.   $$\left\{ \begin{array} { l } 
4 x - 3 y = - 30 \
6 x + 9 y = - 18
\end{array} \right.$$

Accepted Answer

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

Question 8

Find all minors of the matrix.   $\left[ \begin{array} { c c c } 
- 9 & 0 & 6 \
4 & 5 & 8 \
8 & - 8 & 4
\end{array} \right]$

Accepted Answer

A)  $M _ { 11 } = 84 , M _ { 12 } = - 48 , M _ { 13 } = - 72 , M _ { 21 } = 48 , M _ { 22 } = - 84 , M _ { 23 } = 72 , M _ { 31 } = - 30 , M _ { 32 } = - 96 , M _ { 33 } = - 45$ 
B)  $M _ { 11 } = - 72 , M _ { 12 } = 48 , M _ { 13 } = - 84 , M _ { 21 } = 72 , M _ { 22 } = - 30 , M _ { 23 } = - 96 , M _ { 31 } = - 45 , M _ { 32 } = 84 , M _ { 33 } = - 48$ 
C)  $M _ { 11 } = 48 , M _ { 12 } = - 84 , M _ { 13 } = 72 , M _ { 21 } = - 30 , M _ { 22 } = - 96 , M _ { 23 } = - 45 , M _ { 31 } = 84 , M _ { 32 } = - 48 , M _ { 33 } = - 72$ 
D)  $M _ { 11 } = - 48 , M _ { 12 } = - 72 , M _ { 13 } = 48 , M _ { 21 } = - 84 , M _ { 22 } = 72 , M _ { 23 } = - 30 , M _ { 31 } = - 96 , M _ { 32 } = - 45 , M _ { 33 } = 84$ 
E)  $M _ { 11 } = - 30 , M _ { 12 } = - 96 , M _ { 13 } = - 45 , M _ { 21 } = 84 , M _ { 22 } = - 48 , M _ { 23 } = - 72 , M _ { 31 } = 48 , M _ { 32 } = - 84 , M _ { 33 } = 72$ 
A)  $M _ { 11 } = 84 , M _ { 12 } = - 48 , M _ { 13 } = - 72 , M _ { 21 } = 48 , M _ { 22 } = - 84 , M _ { 23 } = 72 , M _ { 31 } = - 30 , M _ { 32 } = - 96 , M _ { 33 } = - 45$ 
B)  $M _ { 11 } = - 72 , M _ { 12 } = 48 , M _ { 13 } = - 84 , M _ { 21 } = 72 , M _ { 22 } = - 30 , M _ { 23 } = - 96 , M _ { 31 } = - 45 , M _ { 32 } = 84 , M _ { 33 } = - 48$ 
C)  $M _ { 11 } = 48 , M _ { 12 } = - 84 , M _ { 13 } = 72 , M _ { 21 } = - 30 , M _ { 22 } = - 96 , M _ { 23 } = - 45 , M _ { 31 } = 84 , M _ { 32 } = - 48 , M _ { 33 } = - 72$ 
D)  $M _ { 11 } = - 48 , M _ { 12 } = - 72 , M _ { 13 } = 48 , M _ { 21 } = - 84 , M _ { 22 } = 72 , M _ { 23 } = - 30 , M _ { 31 } = - 96 , M _ { 32 } = - 45 , M _ { 33 } = 84$ 
E)  $M _ { 11 } = - 30 , M _ { 12 } = - 96 , M _ { 13 } = - 45 , M _ { 21 } = 84 , M _ { 22 } = - 48 , M _ { 23 } = - 72 , M _ { 31 } = 48 , M _ { 32 } = - 84 , M _ { 33 } = 72$

Question 9

A triangular region of farmland has become overrun with deer and will be open to the public for hunting to reduce the population.In order to know how many hunters to allow on the land at one time, you need to know the area of the region in square miles.In order to estimate the area of the region, you travel from the southernmost vertex C north 20 miles then west 24 miles (for vertex B), and from the southernmost vertex C you travel 54 miles north then 7 miles east (for vertex A).Use a graphing utility to approximate the number of square miles of land.

Accepted Answer

A) 578 square miles 
B) 51 square miles 
C) 119 square miles 
D) 429 square miles 
E) 718 square miles 
A) 578 square miles 
B) 51 square miles 
C) 119 square miles 
D) 429 square miles 
E) 718 square miles

Question 10

Find the inverse of the matrix $\left[ \begin{array} { c c c } 
7 & 7 & 7 \
21 & 35 & 28 \
21 & 42 & 35
\end{array} \right]$ .

Accepted Answer

A)   $\frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 1 & - 1 \
- 3 & 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$ 
B)   $\frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 1 & 1 \
0 & - 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$ 
C)   $- 7 \left[ \begin{array} { c c c } 
3 & 1 & 0 \
- 3 & 1 & - 1 \
1 & - 3 & 2
\end{array} \right]$ 
D)   $- \frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 0 & - 1 \
- 3 & 2 & - 1 \
2 & - 3 & 3
\end{array} \right]$ 
E)   $7 \left[ \begin{array} { c c c } 
1 & 1 & 1 \
- 3 & 0 & - 1 \
3 & - 3 & 2
\end{array} \right]$ 
A)   $\frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 1 & - 1 \
- 3 & 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$ 
B)   $\frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 1 & 1 \
0 & - 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$ 
C)   $- 7 \left[ \begin{array} { c c c } 
3 & 1 & 0 \
- 3 & 1 & - 1 \
1 & - 3 & 2
\end{array} \right]$ 
D)   $- \frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 0 & - 1 \
- 3 & 2 & - 1 \
2 & - 3 & 3
\end{array} \right]$ 
E)   $7 \left[ \begin{array} { c c c } 
1 & 1 & 1 \
- 3 & 0 & - 1 \
3 & - 3 & 2
\end{array} \right]$

Question 11

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix $$\left[ \begin{array} { c c c } 
0 & 5 & 1 \
- 2 & 1 & 2 \
2 & - 5 & 0
\end{array} \right]$$ .Show all your work.
​

Accepted Answer

The answer of Write a cryptogram for the message "MERRY...

Question 12

Solve the system of linear equations $\left\{ \begin{array} { l } 
3 x + 6 y = 1 \
6 x + 8 y = 2
\end{array} \right.$ using the inverse matrix $\left[ \begin{array} { c c } 
- \frac { 2 } { 3 } & \frac { 1 } { 2 } \
\frac { 1 } { 2 } & - \frac { 1 } { 4 }
\end{array} \right]$

Accepted Answer

A)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 11 } { 12 } \
0
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 1 } { 3 } \
0
\end{array} \right]$ 
C)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 11 } { 12 } \
\frac { 1 } { 3 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 3 } \
- \frac { 1 } { 2 }
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
- \frac { 1 } { 2 } \
\frac { 1 } { 3 }
\end{array} \right.$ 
A)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 11 } { 12 } \
0
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 1 } { 3 } \
0
\end{array} \right]$ 
C)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 11 } { 12 } \
\frac { 1 } { 3 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 3 } \
- \frac { 1 } { 2 }
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
- \frac { 1 } { 2 } \
\frac { 1 } { 3 }
\end{array} \right.$

Question 13

Solve the system of linear equations.   $\left\{ \begin{array} { l } 
x - 2 y = 3 \
2 x - 3 y = - 4
\end{array} \right.$

Accepted Answer

A)   $( - 17,4 )$ 
B)   $( - 17,10 )$ 
C)   $( - 4 , - 17 )$ 
D)   $( 17,10 )$ 
E)   $( - 17 , - 10 )$ 
A)   $( - 17,4 )$ 
B)   $( - 17,10 )$ 
C)   $( - 4 , - 17 )$ 
D)   $( 17,10 )$ 
E)   $( - 17 , - 10 )$

Question 14

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

Accepted Answer

A)   $\left[ \begin{array} { c c c c } 
1 & 0 & - 1 & - 13 \
0 & 1 & - 2 & 2 \
0 & 0 & 1 & 6
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c c } 
1 & 6 & - 17 & - 13 \
0 & 1 & - 2 & 2 \
0 & 0 & 1 & 6
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c c c } 
1 & 6 & 17 & - 7 \
0 & 1 & - 2 & 2 \
0 & 0 & 1 & 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c c } 
1 & 6 & 17 & - 13 \
0 & 1 & - 2 & 2 \
0 & 0 & 1 & 6
\end{array} \right]$ 
E)   $\left[ \begin{array} { c c c c } 
1 & 0 & - 1 & - 7 \
0 & 1 & - 2 & 2 \
0 & 0 & 1 & 6
\end{array} \right]$ 
A)   $\left[ \begin{array} { c c c c } 
1 & 0 & - 1 & - 13 \
0 & 1 & - 2 & 2 \
0 & 0 & 1 & 6
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c c } 
1 & 6 & - 17 & - 13 \
0 & 1 & - 2 & 2 \
0 & 0 & 1 & 6
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c c c } 
1 & 6 & 17 & - 7 \
0 & 1 & - 2 & 2 \
0 & 0 & 1 & 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c c } 
1 & 6 & 17 & - 13 \
0 & 1 & - 2 & 2 \
0 & 0 & 1 & 6
\end{array} \right]$ 
E)   $\left[ \begin{array} { c c c c } 
1 & 0 & - 1 & - 7 \
0 & 1 & - 2 & 2 \
0 & 0 & 1 & 6
\end{array} \right]$

Question 15

Use a determinant to find the area of the triangle shown below.

Accepted Answer

A) 14 square units 
B)  $\frac { 57 } { 4 }$ square units 
C)  $\frac { 27 } { 2 }$ square units 
D) 12 square units 
E)  $\frac { 85 } { 6 }$ square units 
A) 14 square units 
B)  $\frac { 57 } { 4 }$ square units 
C)  $\frac { 27 } { 2 }$ square units 
D) 12 square units 
E)  $\frac { 85 } { 6 }$ square units

Question 16

Use the matrix capabilities of a graphing utility to find the inverse of the matrix $\frac { 1 } { 9 } \left[ \begin{array} { c c c } 
- 4 & - 5 & 3 \
- 4 & - 8 & 3 \
1 & 2 & 0
\end{array} \right]$ (if it exists).

Accepted Answer

A)   $\left[ \begin{array} { c c c } 
- 9 & 12 & 6 \
0 & - 3 & 0 \
0 & 3 & - 9
\end{array} \right]$ 
B)   $\left[ \begin{array} { c c c } 
- 6 & 6 & 9 \
3 & - 3 & 0 \
0 & 3 & 12
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c c } 
- 6 & 0 & 9 \
0 & - 3 & 0 \
0 & 3 & - 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
3 & 3 & 9 \
0 & - 3 & 0 \
0 & 6 & 12
\end{array} \right]$ 
E)  does not exist 
A)   $\left[ \begin{array} { c c c } 
- 9 & 12 & 6 \
0 & - 3 & 0 \
0 & 3 & - 9
\end{array} \right]$ 
B)   $\left[ \begin{array} { c c c } 
- 6 & 6 & 9 \
3 & - 3 & 0 \
0 & 3 & 12
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c c } 
- 6 & 0 & 9 \
0 & - 3 & 0 \
0 & 3 & - 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
3 & 3 & 9 \
0 & - 3 & 0 \
0 & 6 & 12
\end{array} \right]$ 
E)  does not exist

Question 17

Find the uncoded 1 × 3 row matrices for the message "TWO IF BY LAND" by assigning a number to each letter in the alphabet such as 1 = A, 2 = B, 3 = C and so on (with 0 assigned to a blank space);  
Then encode the message using the encoding matrix $$\left[ \begin{array} { c c c } 
0 & 3 & - 3 \
- 3 & - 3 & 3 \
- 3 & 3 & 0
\end{array} \right]$$ .

Accepted Answer

A) Uncoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
20 & 23 & 15
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 9 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 2 & 25
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 12 & 1
\end{array} \right] } \
\begin{array} { l l l l l l l l l l l l } 
T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}
\end{array} \
{ \left[ \begin{array} { l l l } 
14 & 4 & 0
\end{array} \right] } \
\mathrm { N } \mathrm { D } \
\end{array}$$ Encoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
- 114 & 36 & 9
\end{array} \right] \left[ \begin{array} { l l l } 
- 45 & - 9 & 27
\end{array} \right] \left[ \begin{array} { l l l } 
- 81 & 69 & 6
\end{array} \right] } \
{ \left[ \begin{array} { l l l } 
- 39 & - 33 & 36
\end{array} \right] \left[ \begin{array} { l l l } 
- 12 & 30 & - 30
\end{array} \right] }
\end{array}$$ 
B) Uncoded:   $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
20 & 23 & 15
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 9 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 2 & 25
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 12 & 1
\end{array} \right] } \
\begin{array} { l l l l l l l l l l l l } 
T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}
\end{array} \
{ \left[ \begin{array} { l l l } 
14 & 4 & 0
\end{array} \right] } \
\mathrm { N } \mathrm { D } \
\end{array}$$ Encoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
- 81 & 69 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
- 39 & - 33 & 36
\end{array} \right] \left[ \begin{array} { l l l } 
- 12 & 30 & - 30
\end{array} \right] } \
{ \left[ \begin{array} { l l l l } 
- 114 & 36 & 9
\end{array} \right] \left[ \begin{array} { l l l } 
- 45 & - 9 & 27
\end{array} \right] }
\end{array}$$ 
C) Uncoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
20 & 23 & 15
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 9 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 2 & 25
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 12 & 1
\end{array} \right] } \
\begin{array} { l l l l l l l l l l l l } 
T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}
\end{array} \
{ \left[ \begin{array} { l l l } 
14 & 4 & 0
\end{array} \right] } \
\mathrm { N } \mathrm { D } \
\end{array}$$ Encoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
- 39 & - 33 & 36
\end{array} \right] \left[ \begin{array} { l l l } 
- 12 & 30 & - 30
\end{array} \right] \left[ \begin{array} { l l l l } 
- 114 & 36 & 9
\end{array} \right] } \
{ \left[ \begin{array} { l l l } 
- 45 & - 9 & 27
\end{array} \right] \left[ \begin{array} { l l l } 
- 81 & 69 & 6
\end{array} \right] }
\end{array}$$ 
D) Uncoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
20 & 23 & 15
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 9 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 2 & 25
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 12 & 1
\end{array} \right] } \
\begin{array} { l l l l l l l l l l l l } 
T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}
\end{array} \
{ \left[ \begin{array} { l l l } 
14 & 4 & 0
\end{array} \right] } \
\mathrm { N } \mathrm { D } \
\end{array}$$ Encoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
- 12 & 30 & - 30
\end{array} \right] \left[ \begin{array} { l l l l } 
- 114 & 36 & 9
\end{array} \right] \left[ \begin{array} { l l l } 
- 45 & - 9 & 27
\end{array} \right] } \
{ \left[ \begin{array} { l l l } 
- 81 & 69 & 6
\end{array} \right] \left[ \begin{array} { l l l l } 
- 39 & - 33 & 36
\end{array} \right] }
\end{array}$$ 
E) Uncoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
20 & 23 & 15
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 9 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 2 & 25
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 12 & 1
\end{array} \right] } \
\begin{array} { l l l l l l l l l l l l } 
T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}
\end{array} \
{ \left[ \begin{array} { l l l } 
14 & 4 & 0
\end{array} \right] } \
\mathrm { N } \mathrm { D } \
\end{array}$$ Encoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
- 45 & - 9 & 27
\end{array} \right] \left[ \begin{array} { l l l } 
- 81 & 69 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
- 39 & - 33 & 36
\end{array} \right] } \
{ \left[ \begin{array} { l l l } 
- 12 & 30 & - 30
\end{array} \right] \left[ \begin{array} { l l l } 
- 114 & 36 & 9
\end{array} \right] }
\end{array}$$ 
A) Uncoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
20 & 23 & 15
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 9 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 2 & 25
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 12 & 1
\end{array} \right] } \
\begin{array} { l l l l l l l l l l l l } 
T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}
\end{array} \
{ \left[ \begin{array} { l l l } 
14 & 4 & 0
\end{array} \right] } \
\mathrm { N } \mathrm { D } \
\end{array}$$ Encoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
- 114 & 36 & 9
\end{array} \right] \left[ \begin{array} { l l l } 
- 45 & - 9 & 27
\end{array} \right] \left[ \begin{array} { l l l } 
- 81 & 69 & 6
\end{array} \right] } \
{ \left[ \begin{array} { l l l } 
- 39 & - 33 & 36
\end{array} \right] \left[ \begin{array} { l l l } 
- 12 & 30 & - 30
\end{array} \right] }
\end{array}$$ 
B) Uncoded:   $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
20 & 23 & 15
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 9 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 2 & 25
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 12 & 1
\end{array} \right] } \
\begin{array} { l l l l l l l l l l l l } 
T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}
\end{array} \
{ \left[ \begin{array} { l l l } 
14 & 4 & 0
\end{array} \right] } \
\mathrm { N } \mathrm { D } \
\end{array}$$ Encoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
- 81 & 69 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
- 39 & - 33 & 36
\end{array} \right] \left[ \begin{array} { l l l } 
- 12 & 30 & - 30
\end{array} \right] } \
{ \left[ \begin{array} { l l l l } 
- 114 & 36 & 9
\end{array} \right] \left[ \begin{array} { l l l } 
- 45 & - 9 & 27
\end{array} \right] }
\end{array}$$ 
C) Uncoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
20 & 23 & 15
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 9 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 2 & 25
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 12 & 1
\end{array} \right] } \
\begin{array} { l l l l l l l l l l l l } 
T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}
\end{array} \
{ \left[ \begin{array} { l l l } 
14 & 4 & 0
\end{array} \right] } \
\mathrm { N } \mathrm { D } \
\end{array}$$ Encoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
- 39 & - 33 & 36
\end{array} \right] \left[ \begin{array} { l l l } 
- 12 & 30 & - 30
\end{array} \right] \left[ \begin{array} { l l l l } 
- 114 & 36 & 9
\end{array} \right] } \
{ \left[ \begin{array} { l l l } 
- 45 & - 9 & 27
\end{array} \right] \left[ \begin{array} { l l l } 
- 81 & 69 & 6
\end{array} \right] }
\end{array}$$ 
D) Uncoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
20 & 23 & 15
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 9 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 2 & 25
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 12 & 1
\end{array} \right] } \
\begin{array} { l l l l l l l l l l l l } 
T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}
\end{array} \
{ \left[ \begin{array} { l l l } 
14 & 4 & 0
\end{array} \right] } \
\mathrm { N } \mathrm { D } \
\end{array}$$ Encoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
- 12 & 30 & - 30
\end{array} \right] \left[ \begin{array} { l l l l } 
- 114 & 36 & 9
\end{array} \right] \left[ \begin{array} { l l l } 
- 45 & - 9 & 27
\end{array} \right] } \
{ \left[ \begin{array} { l l l } 
- 81 & 69 & 6
\end{array} \right] \left[ \begin{array} { l l l l } 
- 39 & - 33 & 36
\end{array} \right] }
\end{array}$$ 
E) Uncoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
20 & 23 & 15
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 9 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 2 & 25
\end{array} \right] \left[ \begin{array} { l l l } 
0 & 12 & 1
\end{array} \right] } \
\begin{array} { l l l l l l l l l l l l } 
T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}
\end{array} \
{ \left[ \begin{array} { l l l } 
14 & 4 & 0
\end{array} \right] } \
\mathrm { N } \mathrm { D } \
\end{array}$$ Encoded: $$\begin{array} { l } 
{ \left[ \begin{array} { l l l } 
- 45 & - 9 & 27
\end{array} \right] \left[ \begin{array} { l l l } 
- 81 & 69 & 6
\end{array} \right] \left[ \begin{array} { l l l } 
- 39 & - 33 & 36
\end{array} \right] } \
{ \left[ \begin{array} { l l l } 
- 12 & 30 & - 30
\end{array} \right] \left[ \begin{array} { l l l } 
- 114 & 36 & 9
\end{array} \right] }
\end{array}$$

Question 18

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

Accepted Answer

A)  $- \frac { 1 } { 8 }$ 
B)   $\frac { 7 } { 30 }$ 
C)   $- \frac { 1 } { 10 }$ 
D)   $- \frac { 1 } { 12 }$ 
E)   $\frac { 7 } { 30 }$. 
A)  $- \frac { 1 } { 8 }$ 
B)   $\frac { 7 } { 30 }$ 
C)   $- \frac { 1 } { 10 }$ 
D)   $- \frac { 1 } { 12 }$ 
E)   $\frac { 7 } { 30 }$.

Question 19

Find the determinant of $\left[ \begin{array} { c c c } 
3 & - 2 & 8 \
- 3 & 2 & - 6 \
1 & - 3 & 6
\end{array} \right]$ by the method of expansion by cofactors using row 2.

Accepted Answer

A) 14 
B) -14 
C) 24 
D) 34 
E) 4 
A) 14 
B) -14 
C) 24 
D) 34 
E) 4

Question 20

Find the determinant of $\left[ \begin{array} { c c c } 
0 & 1 & 1 \
2 & - 2 & 0 \
- 1 & - 8 & - 3
\end{array} \right]$ .

Accepted Answer

A) 8 
B) -10 
C) -15 
D) -12 
E) 2 
A) 8 
B) -10 
C) -15 
D) -12 
E) 2

Select the order for the following matrix. $\left[ \begin{array} { c c c c } - 3 & 5 & 10 & 0 \\0 & 0 & 5 & 5 \\4 & 4 & 8 & 5\end{array} \right]$

Find y such that the points are collinear. (5, -4), (3, y), (4, -3)

Find the inverse of A. $A = \left[ \begin{array} { c c c } - 4 & 12 & 4 \\8 & 20 & 0 \\12 & 4 & - 8\end{array} \right]$

Find a value of y such that the triangle with the given vertices has an area of 26 square units. (-6, -6), (2, -2), (-9, y)

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{array} { r l r } 3 x + 2 y - z + w & = 0 \\x - y + 4 z - w & = 28 \\- 2 x + y + 2 z - w & = 3 \\x + y + z + w & = 8\end{array} \right.$

Find the determinant of the matrix. $\left[ \begin{array} { c c } 7 & 1 \\- 1 & 2\end{array} \right]$

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { l } 4 x - 3 y = - 30 \\6 x + 9 y = - 18\end{array} \right.$

Find all minors of the matrix. $\left[ \begin{array} { c c c } - 9 & 0 & 6 \\4 & 5 & 8 \\8 & - 8 & 4\end{array} \right]$

Find the inverse of the matrix $\left[ \begin{array} { c c c } 7 & 7 & 7 \\21 & 35 & 28 \\21 & 42 & 35\end{array} \right]$ .

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix $\left[ \begin{array} { c c c } 0 & 5 & 1 \\- 2 & 1 & 2 \\2 & - 5 & 0\end{array} \right]$ .Show all your work.

Solve the system of linear equations $\left\{ \begin{array} { l } 3 x + 6 y = 1 \\6 x + 8 y = 2\end{array} \right.$ using the inverse matrix $\left[ \begin{array} { c c } - \frac { 2 } { 3 } & \frac { 1 } { 2 } \\\frac { 1 } { 2 } & - \frac { 1 } { 4 }\end{array} \right]$

Solve the system of linear equations. $\left\{ \begin{array} { l } x - 2 y = 3 \\2 x - 3 y = - 4\end{array} \right.$

Use a determinant to find the area of the triangle shown below.

Use the matrix capabilities of a graphing utility to find the inverse of the matrix $\frac { 1 } { 9 } \left[ \begin{array} { c c c } - 4 & - 5 & 3 \\- 4 & - 8 & 3 \\1 & 2 & 0\end{array} \right]$ (if it exists).

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

Find the determinant of $\left[ \begin{array} { c c c } 3 & - 2 & 8 \\- 3 & 2 & - 6 \\1 & - 3 & 6\end{array} \right]$ by the method of expansion by cofactors using row 2.

Find the determinant of $\left[ \begin{array} { c c c } 0 & 1 & 1 \\2 & - 2 & 0 \\- 1 & - 8 & - 3\end{array} \right]$ .

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

Select the order for the following matrix. [−3510000554485]\left[ \begin{array} { c c c c } - 3 & 5 & 10 & 0 \\0 & 0 & 5 & 5 \\4 & 4 & 8 & 5\end{array} \right]​−304​504​1058​055​​

Find y such that the points are collinear. ​ (5, -4), (3, y), (4, -3) ​

Find the inverse of A. A=[−41248200124−8]A = \left[ \begin{array} { c c c } - 4 & 12 & 4 \\8 & 20 & 0 \\12 & 4 & - 8\end{array} \right]A=​−4812​12204​40−8​​

​Find a value of y such that the triangle with the given vertices has an area of 26 square units. ​ ​(-6, -6), ​(2, -2), ​(-9, y) ​

Find the determinant of the matrix. [71−12]\left[ \begin{array} { c c } 7 & 1 \\- 1 & 2\end{array} \right][7−1​12​]

Use Cramer's Rule to solve (if possible) the system of equations. {4x−3y=−306x+9y=−18\left\{ \begin{array} { l } 4 x - 3 y = - 30 \\6 x + 9 y = - 18\end{array} \right.{4x−3y=−306x+9y=−18​

Find all minors of the matrix. [−9064588−84]\left[ \begin{array} { c c c } - 9 & 0 & 6 \\4 & 5 & 8 \\8 & - 8 & 4\end{array} \right]​−948​05−8​684​​

Find the inverse of the matrix [777213528214235]\left[ \begin{array} { c c c } 7 & 7 & 7 \\21 & 35 & 28 \\21 & 42 & 35\end{array} \right]​72121​73542​72835​​ .

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix [051−2122−50]\left[ \begin{array} { c c c } 0 & 5 & 1 \\- 2 & 1 & 2 \\2 & - 5 & 0\end{array} \right]​0−22​51−5​120​​ .Show all your work. ​

Solve the system of linear equations. {x−2y=32x−3y=−4\left\{ \begin{array} { l } x - 2 y = 3 \\2 x - 3 y = - 4\end{array} \right.{x−2y=32x−3y=−4​

Use a determinant to find the area of the triangle shown below.

Use the matrix capabilities of a graphing utility to find the inverse of the matrix 19[−4−53−4−83120]\frac { 1 } { 9 } \left[ \begin{array} { c c c } - 4 & - 5 & 3 \\- 4 & - 8 & 3 \\1 & 2 & 0\end{array} \right]91​​−4−41​−5−82​330​​ (if it exists).

Find the determinant of the matrix [−16−16−1−25]\left[ \begin{array} { c } - \frac { 1 } { 6 } - \frac { 1 } { 6 } \\- 1 - \frac { 2 } { 5 }\end{array} \right][−61​−61​−1−52​​] .

Find the determinant of [3−28−32−61−36]\left[ \begin{array} { c c c } 3 & - 2 & 8 \\- 3 & 2 & - 6 \\1 & - 3 & 6\end{array} \right]​3−31​−22−3​8−66​​ by the method of expansion by cofactors using row 2.

Find the determinant of [0112−20−1−8−3]\left[ \begin{array} { c c c } 0 & 1 & 1 \\2 & - 2 & 0 \\- 1 & - 8 & - 3\end{array} \right]​02−1​1−2−8​10−3​​ .

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

Select the order for the following matrix. $\left[ \begin{array} { c c c c } - 3 & 5 & 10 & 0 \\0 & 0 & 5 & 5 \\4 & 4 & 8 & 5\end{array} \right]$

Find y such that the points are collinear. (5, -4), (3, y), (4, -3)

Find the inverse of A. $A = \left[ \begin{array} { c c c } - 4 & 12 & 4 \\8 & 20 & 0 \\12 & 4 & - 8\end{array} \right]$

Find a value of y such that the triangle with the given vertices has an area of 26 square units. (-6, -6), (2, -2), (-9, y)

Find the determinant of the matrix. $\left[ \begin{array} { c c } 7 & 1 \\- 1 & 2\end{array} \right]$

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { l } 4 x - 3 y = - 30 \\6 x + 9 y = - 18\end{array} \right.$

Find all minors of the matrix. $\left[ \begin{array} { c c c } - 9 & 0 & 6 \\4 & 5 & 8 \\8 & - 8 & 4\end{array} \right]$

Find the inverse of the matrix $\left[ \begin{array} { c c c } 7 & 7 & 7 \\21 & 35 & 28 \\21 & 42 & 35\end{array} \right]$ .

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix $\left[ \begin{array} { c c c } 0 & 5 & 1 \\- 2 & 1 & 2 \\2 & - 5 & 0\end{array} \right]$ .Show all your work.

Solve the system of linear equations. $\left\{ \begin{array} { l } x - 2 y = 3 \\2 x - 3 y = - 4\end{array} \right.$

Use the matrix capabilities of a graphing utility to find the inverse of the matrix $\frac { 1 } { 9 } \left[ \begin{array} { c c c } - 4 & - 5 & 3 \\- 4 & - 8 & 3 \\1 & 2 & 0\end{array} \right]$ (if it exists).

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

Find the determinant of $\left[ \begin{array} { c c c } 3 & - 2 & 8 \\- 3 & 2 & - 6 \\1 & - 3 & 6\end{array} \right]$ by the method of expansion by cofactors using row 2.

Find the determinant of $\left[ \begin{array} { c c c } 0 & 1 & 1 \\2 & - 2 & 0 \\- 1 & - 8 & - 3\end{array} \right]$ .