Question 1

Find 5A.  $$A = \left[ \begin{array} { l l } 
4 & 9 \
4 & 7
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c c } 
20 & 9 \
4 & 7
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
4 & 45 \
20 & 35
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
4 & 45 \
4 & 35
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
20 & 9 \
20 & 35
\end{array} \right]$ 
E)  $\left[ \begin{array} { l l } 
20 & 45 \
20 & 35
\end{array} \right]$ 
A)  $\left[ \begin{array} { c c } 
20 & 9 \
4 & 7
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
4 & 45 \
20 & 35
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
4 & 45 \
4 & 35
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
20 & 9 \
20 & 35
\end{array} \right]$ 
E)  $\left[ \begin{array} { l l } 
20 & 45 \
20 & 35
\end{array} \right]$

Question 2

Use a determinant and the given vertices of a triangle to find the area of the triangle.  
(-2, 5), (3, 3), (-1, 6)

Accepted Answer

A)  $\frac { 7 } { 2 }$ 
B)  $\frac { 2 } { 19 }$ 
C)  $\frac { 3 } { 7 }$ 
D)  $\frac { 6 } { 11 }$ 
E)  $\frac { 5 } { 17 }$ 
A)  $\frac { 7 } { 2 }$ 
B)  $\frac { 2 } { 19 }$ 
C)  $\frac { 3 } { 7 }$ 
D)  $\frac { 6 } { 11 }$ 
E)  $\frac { 5 } { 17 }$

Question 3

Use a determinant and the given vertices of a triangle to find the area of the triangle.

Accepted Answer

A) 52 
B)  $\frac { 4 } { 19 }$ 
C) 25 
D)  $\frac { 2 } { 17 }$ 
E) 54 
A) 52 
B)  $\frac { 4 } { 19 }$ 
C) 25 
D)  $\frac { 2 } { 17 }$ 
E) 54

Question 4

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix, by assigning a number to each letter in the alphabet (with 0 assigned to a blank space, 0 = _, 1 = A, 2 = B and so on.)   $\left[ \begin{array} { c c c } 
0 & 5 & 1 \
- 1 & 1 & 1 \
1 & - 5 & 0
\end{array} \right]$ .

Accepted Answer

A) $\begin{array} { c c c c c c c c c } 
1 & - 36 & 28 & 18 & - 29 & 14 & 13 & - 20 & 18 \
- 25 & 115 & 43 & 10 & - 67 & 11 & & &
\end{array}$ 
B) $\begin{array} { c c c c c c c c c } 
13 & - 20 & 18 & - 25 & 115 & 43 & 10 & - 67 & 11 \
1 & - 36 & 28 & 18 & - 29 & 14 & & &
\end{array}$ 
C) $\begin{array} { r r r l l l l l l } 
10 & - 67 & 11 & 1 & - 36 & 28 & 18 & - 29 & 14 \
13 & - 20 & 18 & - 25 & 115 & 43 & & &
\end{array}$ 
D) $\begin{array} { c c c c c c c c c } 
- 25 & 115 & 43 & 10 & - 67 & 11 & 1 & - 36 & 28 \
18 & - 29 & 14 & 13 & - 20 & 18 & & &
\end{array}$ 
E) $\begin{array} { c c c c c c c c c } 
18 & - 29 & 14 & 13 & - 20 & 18 & - 25 & 115 & 43 \
10 & - 67 & 11 & 1 & - 36 & 28 & & &
\end{array}$ 
A) $\begin{array} { c c c c c c c c c } 
1 & - 36 & 28 & 18 & - 29 & 14 & 13 & - 20 & 18 \
- 25 & 115 & 43 & 10 & - 67 & 11 & & &
\end{array}$ 
B) $\begin{array} { c c c c c c c c c } 
13 & - 20 & 18 & - 25 & 115 & 43 & 10 & - 67 & 11 \
1 & - 36 & 28 & 18 & - 29 & 14 & & &
\end{array}$ 
C) $\begin{array} { r r r l l l l l l } 
10 & - 67 & 11 & 1 & - 36 & 28 & 18 & - 29 & 14 \
13 & - 20 & 18 & - 25 & 115 & 43 & & &
\end{array}$ 
D) $\begin{array} { c c c c c c c c c } 
- 25 & 115 & 43 & 10 & - 67 & 11 & 1 & - 36 & 28 \
18 & - 29 & 14 & 13 & - 20 & 18 & & &
\end{array}$ 
E) $\begin{array} { c c c c c c c c c } 
18 & - 29 & 14 & 13 & - 20 & 18 & - 25 & 115 & 43 \
10 & - 67 & 11 & 1 & - 36 & 28 & & &
\end{array}$

Question 5

Use Cramer's Rule to solve (if possible) the system of equations.   $$\left\{ \begin{array} { r } 
- 0.4 x + 0.8 y = 5.52 \
0.2 x + 0.3 y = 2.63
\end{array} \right.$$

Accepted Answer

A)  $\left( \frac { 77 } { 10 } , \frac { 8 } { 5 } \right)$ 
B)  $\left( \frac { 8 } { 5 } , \frac { 77 } { 10 } \right)$ 
C)  $\left( - \frac { 8 } { 5 } , - \frac { 77 } { 10 } \right)$ 
D)  $\left( \frac { 8 } { 5 } , - \frac { 77 } { 10 } \right)$ 
E)  $\left( - \frac { 8 } { 5 } , \frac { 77 } { 10 } \right)$ 
A)  $\left( \frac { 77 } { 10 } , \frac { 8 } { 5 } \right)$ 
B)  $\left( \frac { 8 } { 5 } , \frac { 77 } { 10 } \right)$ 
C)  $\left( - \frac { 8 } { 5 } , - \frac { 77 } { 10 } \right)$ 
D)  $\left( \frac { 8 } { 5 } , - \frac { 77 } { 10 } \right)$ 
E)  $\left( - \frac { 8 } { 5 } , \frac { 77 } { 10 } \right)$

Question 6

Solve the system of linear equations   $\left\{ \begin{array} { l l } 
7 x + 7 y + 7 z & = 0 \
21 x + 35 y + 28 z & = 6 \
21 x + 42 y + 35 z & = 1
\end{array} \right.$  using the inverse matrix $\frac { 1 } { 7 } \left[ \begin{array} { c c c } 
1 & 1 & - 1 \
- 3 & 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$ .

Accepted Answer

A)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 5 } { 7 } \
\frac { 11 } { 7 } \
\frac { - 12 } { 7 }
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 5 } { 7 } \
\frac { 11 } { 7 } \
\frac { - 16 } { 7 }
\end{array} \right]$ 
C)  $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 16 } { 7 } \
\frac { 5 } { 7 } \
\frac { 11 } { 7 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 12 } { 7 } \
\frac { 11 } { 7 } \
\frac { - 5 } { 7 }
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 5 } { 7 } \
\frac { - 16 } { 7 } \
\frac { 11 } { 7 }
\end{array} \right]$ 
A)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 5 } { 7 } \
\frac { 11 } { 7 } \
\frac { - 12 } { 7 }
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 5 } { 7 } \
\frac { 11 } { 7 } \
\frac { - 16 } { 7 }
\end{array} \right]$ 
C)  $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 16 } { 7 } \
\frac { 5 } { 7 } \
\frac { 11 } { 7 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 12 } { 7 } \
\frac { 11 } { 7 } \
\frac { - 5 } { 7 }
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 5 } { 7 } \
\frac { - 16 } { 7 } \
\frac { 11 } { 7 }
\end{array} \right]$

Question 7

Use a graphing calculator to find the inverse of the matrix.  $\left[ \begin{array} { l l l l } 
1 & 4 & 8 & 3 \
0 & 1 & 4 & 8 \
0 & 0 & 1 & 4 \
0 & 0 & 0 & 1
\end{array} \right]$

Accepted Answer

A)   $\left[ \begin{array} { c c c c } 
1 & 4 & 8 & - 3 \
0 & 1 & - 4 & 8 \
0 & 0 & 1 & 4 \
0 & 0 & 0 & 1
\end{array} \right]$ 
B)   $\left[ \begin{array} { c c c c } 
1 & - 4 & 8 & 3 \
0 & 1 & - 4 & 8 \
0 & 0 & 1 & - 4 \
0 & 0 & 0 & 1
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c c c } 
1 & - 4 & - 8 & - 3 \
0 & 1 & - 4 & - 8 \
0 & 0 & 1 & - 4 \
0 & 0 & 0 & 1
\end{array} \right]$ 
D)   $\left[ \begin{array} { c c c c } 
1 & - 4 & 8 & - 3 \
0 & 1 & - 4 & 8 \
0 & 0 & 1 & - 4 \
0 & 0 & 0 & 1
\end{array} \right]$ 
E)   $\left[ \begin{array} { c c c c } 
1 & 4 & 8 & - 3 \
0 & 1 & 4 & 8 \
0 & 0 & 1 & 4 \
0 & 0 & 0 & 1
\end{array} \right]$ 
A)   $\left[ \begin{array} { c c c c } 
1 & 4 & 8 & - 3 \
0 & 1 & - 4 & 8 \
0 & 0 & 1 & 4 \
0 & 0 & 0 & 1
\end{array} \right]$ 
B)   $\left[ \begin{array} { c c c c } 
1 & - 4 & 8 & 3 \
0 & 1 & - 4 & 8 \
0 & 0 & 1 & - 4 \
0 & 0 & 0 & 1
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c c c } 
1 & - 4 & - 8 & - 3 \
0 & 1 & - 4 & - 8 \
0 & 0 & 1 & - 4 \
0 & 0 & 0 & 1
\end{array} \right]$ 
D)   $\left[ \begin{array} { c c c c } 
1 & - 4 & 8 & - 3 \
0 & 1 & - 4 & 8 \
0 & 0 & 1 & - 4 \
0 & 0 & 0 & 1
\end{array} \right]$ 
E)   $\left[ \begin{array} { c c c c } 
1 & 4 & 8 & - 3 \
0 & 1 & 4 & 8 \
0 & 0 & 1 & 4 \
0 & 0 & 0 & 1
\end{array} \right]$

Question 8

Solve for X in the equation given. $$3 X = - 5 A - B , A = \left[ \begin{array} { c c } 
9 & 1 \
- 3 & - 8
\end{array} \right] \text { and } B = \left[ \begin{array} { c c } 
- 33 & 4 \
30 & 46
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { l l } 
- 4 & - 3 \
- 5 & - 2
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
- 4 & 3 \
5 & - 2
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
- 12 & - 9 \
- 15 & - 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { l l } 
1 & 0 \
0 & 1
\end{array} \right]$ 
E)  $\left[ \begin{array} { l l } 
- 4 & 0 \
- 5 & 4
\end{array} \right]$ 
A)  $\left[ \begin{array} { l l } 
- 4 & - 3 \
- 5 & - 2
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
- 4 & 3 \
5 & - 2
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
- 12 & - 9 \
- 15 & - 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { l l } 
1 & 0 \
0 & 1
\end{array} \right]$ 
E)  $\left[ \begin{array} { l l } 
- 4 & 0 \
- 5 & 4
\end{array} \right]$

Question 9

Find the determinant of the matrix.   $\left[ \begin{array} { c c } 
2 & - 5 \
0 & 0
\end{array} \right]$

Accepted Answer

A)  $7$ 
B) 0 
C)  $- 10$ 
D) 10 
E)  $- 7$ 
A)  $7$ 
B) 0 
C)  $- 10$ 
D) 10 
E)  $- 7$

Question 10

Solve the system of linear equations   $\left\{ \begin{array} { l l } 
9 x + 9 y + 9 z & = 1 \
27 x + 45 y + 36 z & = - 3 \
27 x + 54 y + 45 z & = 2
\end{array} \right.$  using the inverse matrix $\frac { 1 } { 9 } \left[ \begin{array} { c c c } 
1 & 1 & - 1 \
- 3 & 2 & - 1 \
3 & - 3 & 2
\end{array} \right]$ .

Accepted Answer

A)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 5 } { 9 } \
\frac { - 2 } { 3 } \
\frac { 4 } { 9 }
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 4 } { 9 } \
\frac { - 11 } { 9 } \
\frac { 16 } { 9 }
\end{array} \right]$ 
C)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 11 } { 9 } \
\frac { 16 } { 9 } \
\frac { - 4 } { 9 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 5 } { 9 } \
\frac { 1 } { 3 } \
\frac { - 2 } { 9 }
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 16 } { 9 } \
\frac { - 4 } { 9 } \
\frac { - 11 } { 9 }
\end{array} \right]$ 
A)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 5 } { 9 } \
\frac { - 2 } { 3 } \
\frac { 4 } { 9 }
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 4 } { 9 } \
\frac { - 11 } { 9 } \
\frac { 16 } { 9 }
\end{array} \right]$ 
C)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 11 } { 9 } \
\frac { 16 } { 9 } \
\frac { - 4 } { 9 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 5 } { 9 } \
\frac { 1 } { 3 } \
\frac { - 2 } { 9 }
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 16 } { 9 } \
\frac { - 4 } { 9 } \
\frac { - 11 } { 9 }
\end{array} \right]$

Question 11

A large region of forest has been infested with gypsy moths.The region is roughly triangular, as shown in the figure on the next page.From the northernmost vertex A of the region, the distances to the other vertices are x = 25 miles south and 10 miles east (for vertex B), and 20 miles south and 28 miles east (for vertex C).Use a graphing utility to approximate the number of square miles in this region. ​   ​

Accepted Answer

A) 250 mi2 
B) 280 mi2 
C) 260 mi2 
D) 290 mi2 
E) 270 mi2 
A) 250 mi2 
B) 280 mi2 
C) 260 mi2 
D) 290 mi2 
E) 270 mi2

Question 12

Determine whether the matrix is in row-echelon form.If it is, determine if it is also in reduced row-echelon form. $\left[ \begin{array} { r r r r } 
1 & 8 & - 8 & 4 \
0 & 1 & 0 & 6 \
0 & 0 & 1 & - 9
\end{array} \right]$

Accepted Answer

A) row-echelon form and reduced row-echelon form 
B) row-echelon form 
C) neither 
A) row-echelon form and reduced row-echelon form 
B) row-echelon form 
C) neither

Question 13

If possible, find A - B. $$A = \left[ \begin{array} { c c } 
- 9 & 5 \
3 & 2
\end{array} \right] , B = \left[ \begin{array} { c c } 
- 1 & 4 \
7 & 3
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c c } 
- 8 & 1 \
- 4 & - 1
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
- 8 & 5 \
3 & - 1
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
- 26 & - 51 \
- 17 & 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
- 10 & 9 \
10 & 5
\end{array} \right]$ 
E) not possible 
A)  $\left[ \begin{array} { c c } 
- 8 & 1 \
- 4 & - 1
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
- 8 & 5 \
3 & - 1
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
- 26 & - 51 \
- 17 & 6
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
- 10 & 9 \
10 & 5
\end{array} \right]$ 
E) not possible

Question 14

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

Accepted Answer

A)  $\left[ \begin{array} { c c c } 
- 8 & 15 & 0 \
0 & - 6 & - 18 \
18 & 24 & - 3 \
15 & 18 & 12 \
- 6 & - 3 & 0
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
- 6 & 15 & 0 \
15 & - 6 & 12 \
18 & 24 & - 3 \
0 & 18 & - 18 \
- 6 & - 3 & 0
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
- 10 & 15 & 0 \
15 & - 6 & 12 \
- 6 & 24 & 0 \
0 & 18 & - 18 \
18 & - 3 & - 3
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 4 & 15 & 0 \
15 & - 6 & 12 \
- 3 & 24 & 18 \
0 & 18 & - 18 \
0 & - 3 & - 6
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c c } 
- 2 & 15 & 0 \
12 & - 6 & 15 \
18 & 24 & - 3 \
- 18 & 18 & 0 \
- 6 & - 3 & 0
\end{array} \right]$ 
A)  $\left[ \begin{array} { c c c } 
- 8 & 15 & 0 \
0 & - 6 & - 18 \
18 & 24 & - 3 \
15 & 18 & 12 \
- 6 & - 3 & 0
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
- 6 & 15 & 0 \
15 & - 6 & 12 \
18 & 24 & - 3 \
0 & 18 & - 18 \
- 6 & - 3 & 0
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
- 10 & 15 & 0 \
15 & - 6 & 12 \
- 6 & 24 & 0 \
0 & 18 & - 18 \
18 & - 3 & - 3
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
- 4 & 15 & 0 \
15 & - 6 & 12 \
- 3 & 24 & 18 \
0 & 18 & - 18 \
0 & - 3 & - 6
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c c } 
- 2 & 15 & 0 \
12 & - 6 & 15 \
18 & 24 & - 3 \
- 18 & 18 & 0 \
- 6 & - 3 & 0
\end{array} \right]$

Question 15

Find the inverse of the matrix (if it exists).   $\left[ \begin{array} { l l } 
6 & - 7 \
7 & - 8
\end{array} \right]$

Accepted Answer

A)   $\left[ \begin{array} { c c } 
- 8 & 7 \
7 & 6
\end{array} \right]$ 
B)   $\left[ \begin{array} { c c } 
- 8 & 7 \
- 7 & 6
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c } 
- 8 & - 7 \
- 7 & 6
\end{array} \right]$ 
D)   $\left[ \begin{array} { c c } 
- 8 & 7 \
7 & - 6
\end{array} \right]$ 
E)   $\left[ \begin{array} { c c } 
- 8 & - 7 \
7 & 6
\end{array} \right]$ 
A)   $\left[ \begin{array} { c c } 
- 8 & 7 \
7 & 6
\end{array} \right]$ 
B)   $\left[ \begin{array} { c c } 
- 8 & 7 \
- 7 & 6
\end{array} \right]$ 
C)   $\left[ \begin{array} { c c } 
- 8 & - 7 \
- 7 & 6
\end{array} \right]$ 
D)   $\left[ \begin{array} { c c } 
- 8 & 7 \
7 & - 6
\end{array} \right]$ 
E)   $\left[ \begin{array} { c c } 
- 8 & - 7 \
7 & 6
\end{array} \right]$

Question 16

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $\left[ \begin{array} { c c c c c } 
- 1 & 0 & 0 & 0 & 0 \
0 & 2 & 0 & 0 & 0 \
0 & 0 & - 3 & 0 & 0 \
0 & 0 & 0 & 1 & 0 \
0 & 0 & 0 & 0 & - 2
\end{array} \right]$ .

Accepted Answer

A) -6 
B) 4 
C) -2 
D) -1 
E) -12 
A) -6 
B) 4 
C) -2 
D) -1 
E) -12

Question 17

Use a determinant and the given vertices of a triangle to find the area of the triangle.  
(-2, 4), (2, 5), (6, -4)

Accepted Answer

A)  $\frac { 6 } { 19 }$ 
B) 20 
C)  $\frac { 5 } { 17 }$ 
D) -20 
E) 40 
A)  $\frac { 6 } { 19 }$ 
B) 20 
C)  $\frac { 5 } { 17 }$ 
D) -20 
E) 40

Question 18

Solve the system of linear equations $\left\{ \begin{array} { l l } 
- 6 x + 18 y + 6 z & = 1 \
12 x + 30 y & = 2 \
18 x + 6 y - 12 z & = - 1
\end{array} \right.$ using an inverse matrix.

Accepted Answer

A)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 6 } \
\frac { - 1 } { 2 } \
\frac { 1 } { 3 }
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 1 } { 6 } \
\frac { - 1 } { 2 } \
\frac { - 1 } { 3 }
\end{array} \right]$ 
C)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 1 } { 6 } \
0 \
\frac { 1 } { 2 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
0 \
\frac { 1 } { 3 } \
\frac { - 1 } { 6 }
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 1 } { 6 } \
0 \
\frac { 1 } { 3 }
\end{array} \right]$ 
A)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 1 } { 6 } \
\frac { - 1 } { 2 } \
\frac { 1 } { 3 }
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 1 } { 6 } \
\frac { - 1 } { 2 } \
\frac { - 1 } { 3 }
\end{array} \right]$ 
C)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 1 } { 6 } \
0 \
\frac { 1 } { 2 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
0 \
\frac { 1 } { 3 } \
\frac { - 1 } { 6 }
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 1 } { 6 } \
0 \
\frac { 1 } { 3 }
\end{array} \right]$

Question 19

An augmented matrix that represents a system of linear equations (in variables x, y, and z) has been reduced using Gauss-Jordan elimination.Write the solution represented by the augmented matrix. $$\left[ \begin{array} { r r r r r } 
1 & 0&0 &  \vdots&2 \
0 & 1 & 0 & \vdots & 3 \
0 & 0 & 1 & \vdots &4
\end{array} \right]$$

Accepted Answer

A) x = 0, y = 0, z = 0 
B) x = 2x, y = 4y, z = 3z 
C) x = 2, y = 3, z = 4 
D) x = -2, y = -3, z = -4 
E) x = 2, y = 0, z = 0 
A) x = 0, y = 0, z = 0 
B) x = 2x, y = 4y, z = 3z 
C) x = 2, y = 3, z = 4 
D) x = -2, y = -3, z = -4 
E) x = 2, y = 0, z = 0

Question 20

Perform the sequence of row operations on the following matrix. Add R3 to R4.
  $\left[ \begin{array} { c c } 
7 & 1 \
0 & 2 \
- 3 & 4 \
4 & 1
\end{array} \right]$

Accepted Answer

A)   $\left[ \begin{array} { l l } 
7 & 1 \
0 & 2 \
7 & 5 \
4 & 1
\end{array} \right]$ 
B)   $\left[ \begin{array} { c c } 
7 & 1 \
0 & 2 \
- 3 & 4 \
11 & 7
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
7 & 1 \
0 & 2 \
1 & 5 \
4 & 1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
7 & 1 \
0 & 2 \
- 3 & 4 \
7 & 11
\end{array} \right]$ 
E)   $\left[ \begin{array} { c c } 
7 & 1 \
0 & 2 \
- 3 & 4 \
1 & 5
\end{array} \right]$ 
A)   $\left[ \begin{array} { l l } 
7 & 1 \
0 & 2 \
7 & 5 \
4 & 1
\end{array} \right]$ 
B)   $\left[ \begin{array} { c c } 
7 & 1 \
0 & 2 \
- 3 & 4 \
11 & 7
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
7 & 1 \
0 & 2 \
1 & 5 \
4 & 1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
7 & 1 \
0 & 2 \
- 3 & 4 \
7 & 11
\end{array} \right]$ 
E)   $\left[ \begin{array} { c c } 
7 & 1 \
0 & 2 \
- 3 & 4 \
1 & 5
\end{array} \right]$

Find 5A. $A = \left[ \begin{array} { l l } 4 & 9 \\4 & 7\end{array} \right]$

Use a determinant and the given vertices of a triangle to find the area of the triangle. (-2, 5), (3, 3), (-1, 6)

Use a determinant and the given vertices of a triangle to find the area of the triangle.

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix, by assigning a number to each letter in the alphabet (with 0 assigned to a blank space, 0 = _, 1 = A, 2 = B and so on.) $\left[ \begin{array} { c c c } 0 & 5 & 1 \\- 1 & 1 & 1 \\1 & - 5 & 0\end{array} \right]$ .

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { r } - 0.4 x + 0.8 y = 5.52 \\0.2 x + 0.3 y = 2.63\end{array} \right.$

Use a graphing calculator to find the inverse of the matrix. $\left[ \begin{array} { l l l l } 1 & 4 & 8 & 3 \\0 & 1 & 4 & 8 \\0 & 0 & 1 & 4 \\0 & 0 & 0 & 1\end{array} \right]$

Solve for X in the equation given. $3 X = - 5 A - B , A = \left[ \begin{array} { c c } 9 & 1 \\- 3 & - 8\end{array} \right] \text { and } B = \left[ \begin{array} { c c } - 33 & 4 \\30 & 46\end{array} \right]$

Find the determinant of the matrix. $\left[ \begin{array} { c c } 2 & - 5 \\0 & 0\end{array} \right]$

Determine whether the matrix is in row-echelon form.If it is, determine if it is also in reduced row-echelon form. $\left[ \begin{array} { r r r r } 1 & 8 & - 8 & 4 \\0 & 1 & 0 & 6 \\0 & 0 & 1 & - 9\end{array} \right]$

If possible, find A - B. $A = \left[ \begin{array} { c c } - 9 & 5 \\3 & 2\end{array} \right] , B = \left[ \begin{array} { c c } - 1 & 4 \\7 & 3\end{array} \right]$

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

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { l l } 6 & - 7 \\7 & - 8\end{array} \right]$

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $\left[ \begin{array} { c c c c c } - 1 & 0 & 0 & 0 & 0 \\0 & 2 & 0 & 0 & 0 \\0 & 0 & - 3 & 0 & 0 \\0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & - 2\end{array} \right]$ .

Use a determinant and the given vertices of a triangle to find the area of the triangle. (-2, 4), (2, 5), (6, -4)

Solve the system of linear equations $\left\{ \begin{array} { l l } - 6 x + 18 y + 6 z & = 1 \\12 x + 30 y & = 2 \\18 x + 6 y - 12 z & = - 1\end{array} \right.$ using an inverse matrix.

Perform the sequence of row operations on the following matrix. Add R₃to R₄. $\left[ \begin{array} { c c } 7 & 1 \\0 & 2 \\- 3 & 4 \\4 & 1\end{array} \right]$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Exam 8: Matrices and Determinants

Find 5A. A=[4947]A = \left[ \begin{array} { l l } 4 & 9 \\4 & 7\end{array} \right]A=[44​97​]

Use a determinant and the given vertices of a triangle to find the area of the triangle. (-2, 5), (3, 3), (-1, 6)

Use a determinant and the given vertices of a triangle to find the area of the triangle.

Use Cramer's Rule to solve (if possible) the system of equations. {−0.4x+0.8y=5.520.2x+0.3y=2.63\left\{ \begin{array} { r } - 0.4 x + 0.8 y = 5.52 \\0.2 x + 0.3 y = 2.63\end{array} \right.{−0.4x+0.8y=5.520.2x+0.3y=2.63​

Use a graphing calculator to find the inverse of the matrix. [1483014800140001]\left[ \begin{array} { l l l l } 1 & 4 & 8 & 3 \\0 & 1 & 4 & 8 \\0 & 0 & 1 & 4 \\0 & 0 & 0 & 1\end{array} \right]​1000​4100​8410​3841​​

Find the determinant of the matrix. [2−500]\left[ \begin{array} { c c } 2 & - 5 \\0 & 0\end{array} \right][20​−50​]

Determine whether the matrix is in row-echelon form.If it is, determine if it is also in reduced row-echelon form. [18−840106001−9]\left[ \begin{array} { r r r r } 1 & 8 & - 8 & 4 \\0 & 1 & 0 & 6 \\0 & 0 & 1 & - 9\end{array} \right]​100​810​−801​46−9​​

If possible, find A - B. A=[−9532],B=[−1473]A = \left[ \begin{array} { c c } - 9 & 5 \\3 & 2\end{array} \right] , B = \left[ \begin{array} { c c } - 1 & 4 \\7 & 3\end{array} \right]A=[−93​52​],B=[−17​43​]

Find 3A. A=[−2505−2468−106−6−2−10]A = \left[ \begin{array} { c c c } - 2 & 5 & 0 \\5 & - 2 & 4 \\6 & 8 & - 1 \\0 & 6 & - 6 \\- 2 & - 1 & 0\end{array} \right]A=​−2560−2​5−286−1​04−1−60​​

Find the inverse of the matrix (if it exists). [6−77−8]\left[ \begin{array} { l l } 6 & - 7 \\7 & - 8\end{array} \right][67​−7−8​]

Use a determinant and the given vertices of a triangle to find the area of the triangle. (-2, 4), (2, 5), (6, -4)

Solve the system of linear equations {−6x+18y+6z=112x+30y=218x+6y−12z=−1\left\{ \begin{array} { l l } - 6 x + 18 y + 6 z & = 1 \\12 x + 30 y & = 2 \\18 x + 6 y - 12 z & = - 1\end{array} \right.⎩⎨⎧​−6x+18y+6z12x+30y18x+6y−12z​=1=2=−1​ using an inverse matrix.

Perform the sequence of row operations on the following matrix. Add R3 to R4. [7102−3441]\left[ \begin{array} { c c } 7 & 1 \\0 & 2 \\- 3 & 4 \\4 & 1\end{array} \right]​70−34​1241​​

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 5A. $A = \left[ \begin{array} { l l } 4 & 9 \\4 & 7\end{array} \right]$

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { r } - 0.4 x + 0.8 y = 5.52 \\0.2 x + 0.3 y = 2.63\end{array} \right.$

Use a graphing calculator to find the inverse of the matrix. $\left[ \begin{array} { l l l l } 1 & 4 & 8 & 3 \\0 & 1 & 4 & 8 \\0 & 0 & 1 & 4 \\0 & 0 & 0 & 1\end{array} \right]$

Find the determinant of the matrix. $\left[ \begin{array} { c c } 2 & - 5 \\0 & 0\end{array} \right]$

Determine whether the matrix is in row-echelon form.If it is, determine if it is also in reduced row-echelon form. $\left[ \begin{array} { r r r r } 1 & 8 & - 8 & 4 \\0 & 1 & 0 & 6 \\0 & 0 & 1 & - 9\end{array} \right]$

If possible, find A - B. $A = \left[ \begin{array} { c c } - 9 & 5 \\3 & 2\end{array} \right] , B = \left[ \begin{array} { c c } - 1 & 4 \\7 & 3\end{array} \right]$

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

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { l l } 6 & - 7 \\7 & - 8\end{array} \right]$

Solve the system of linear equations $\left\{ \begin{array} { l l } - 6 x + 18 y + 6 z & = 1 \\12 x + 30 y & = 2 \\18 x + 6 y - 12 z & = - 1\end{array} \right.$ using an inverse matrix.

Perform the sequence of row operations on the following matrix. Add R₃to R₄. $\left[ \begin{array} { c c } 7 & 1 \\0 & 2 \\- 3 & 4 \\4 & 1\end{array} \right]$