Question 1

Find a value of y such that the triangle with the given vertices has an area of 8 square units. ​
​(5, 6), ​(5, 8), ​(-3, y)
​

Accepted Answer

A) y = 10 or y = -9 
B) y = -10 or y = -9 
C) y = 10 or y = 9 
D) y = 10 or y = 0 
E) y = -10 or y = 9 
A) y = 10 or y = -9 
B) y = -10 or y = -9 
C) y = 10 or y = 9 
D) y = 10 or y = 0 
E) y = -10 or y = 9

Question 2

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$\left\{ \begin{aligned}
7 x - 6 y & = - 40 \
- 5 x + 7 y & = 34
\end{aligned} \right.$$

Accepted Answer

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

Question 3

Evaluate the expression. $$\left[ \begin{array} { l l } 
8 & 5 \
6 & 2
\end{array} \right] + \left[ \begin{array} { l c } 
- 5 & - 6 \
- 8 & 2
\end{array} \right] + \left[ \begin{array} { c c } 
7 & - 5 \
5 & 8
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c c } 
- 4 & 4 \
- 7 & - 4
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
3 & - 1 \
- 2 & 4
\end{array} \right]$ 
C) not possible 
D)  $\left[ \begin{array} { l l } 
1 & 0 \
0 & 1
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c } 
10 & - 6 \
3 & 12
\end{array} \right]$ 
A)  $\left[ \begin{array} { c c } 
- 4 & 4 \
- 7 & - 4
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
3 & - 1 \
- 2 & 4
\end{array} \right]$ 
C) not possible 
D)  $\left[ \begin{array} { l l } 
1 & 0 \
0 & 1
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c } 
10 & - 6 \
3 & 12
\end{array} \right]$

Question 4

Solve the system of linear equations $\left\{ \begin{array} { l } 
4 x + 4 y = 3 \
8 x - 20 y = - 9
\end{array} \right.$ using an inverse matrix.

Accepted Answer

A)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 3 } { 14 } \
\frac { 15 } { 28 }
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 9 } { 28 } \
\frac { 12 } { 7 }
\end{array} \right]$ 
C)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 28 } \
\frac { - 3 } { 14 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 7 } \
\frac { 15 } { 28 }
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 9 } { 7 } \
\frac { - 3 } { 14 }
\end{array} \right]$ 
A)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 3 } { 14 } \
\frac { 15 } { 28 }
\end{array} \right]$ 
B)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
\frac { 9 } { 28 } \
\frac { 12 } { 7 }
\end{array} \right]$ 
C)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 28 } \
\frac { - 3 } { 14 }
\end{array} \right]$ 
D)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { 9 } { 7 } \
\frac { 15 } { 28 }
\end{array} \right]$ 
E)   $\left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { c } 
\frac { - 9 } { 7 } \
\frac { - 3 } { 14 }
\end{array} \right]$

Question 5

Find the inverse of the matrix $\left[ \begin{array} { c c } 
- 8 & 14 \
- 4 & 7
\end{array} \right]$ (if it exists).

Accepted Answer

A)   $\frac { 1 } { 4 } \left[ \begin{array} { l l } 
7 & - 14 \
4 & - 8
\end{array} \right]$ 
B)   $- \frac { 1 } { 8 } \left[ \begin{array} { c c } 
7 & - 14 \
4 & - 8
\end{array} \right]$ 
C)   $- \frac { 1 } { 4 } \left[ \begin{array} { c c } 
- 7 & - 14 \
4 & 8
\end{array} \right]$ 
D)   $\frac { 1 } { 7 } \left[ \begin{array} { c c } 
- 8 & 14 \
- 4 & 7
\end{array} \right]$ 
E) does not exist 
A)   $\frac { 1 } { 4 } \left[ \begin{array} { l l } 
7 & - 14 \
4 & - 8
\end{array} \right]$ 
B)   $- \frac { 1 } { 8 } \left[ \begin{array} { c c } 
7 & - 14 \
4 & - 8
\end{array} \right]$ 
C)   $- \frac { 1 } { 4 } \left[ \begin{array} { c c } 
- 7 & - 14 \
4 & 8
\end{array} \right]$ 
D)   $\frac { 1 } { 7 } \left[ \begin{array} { c c } 
- 8 & 14 \
- 4 & 7
\end{array} \right]$ 
E) does not exist

Question 6

Find A + B.   $$A = \left[ \begin{array} { l l } 
5 & - 5 \
2 & - 5
\end{array} \right] , B = \left[ \begin{array} { c c } 
2 & - 5 \
- 5 & 7
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c c } 
7 & - 10 \
- 3 & 2
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
9 & 2 \
- 3 & - 10
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
11 & - 10 \
2 & - 3
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
5 & - 10 \
- 3 & 2
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c } 
3 & - 10 \
- 3 & 2
\end{array} \right]$ 
A)  $\left[ \begin{array} { c c } 
7 & - 10 \
- 3 & 2
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
9 & 2 \
- 3 & - 10
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
11 & - 10 \
2 & - 3
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
5 & - 10 \
- 3 & 2
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c } 
3 & - 10 \
- 3 & 2
\end{array} \right]$

Question 7

Write the matrix in reduced row-echelon form. $$\left[ \begin{array} { r r r } 
4 & 5 & 11 \
3 & 1 & - 11 \
3 & 5 & 17
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { r r r } 
1 & 0 & - 6 \
0 & 0 & 0 \
0 & 0 & 0
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l l } 
1 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 0
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
1 & 0 & - 4 \
0 & 1 & 5 \
0 & 0 & 1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
1 & 0 & - 7 \
0 & 1 & - 6 \
0 & 0 & 0
\end{array} \right]$ 
E)  $\left[ \begin{array} { r r r } 
1 & 0 & - 6 \
0 & 1 & 7 \
0 & 0 & 0
\end{array} \right]$ 
A)  $\left[ \begin{array} { r r r } 
1 & 0 & - 6 \
0 & 0 & 0 \
0 & 0 & 0
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l l } 
1 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 0
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
1 & 0 & - 4 \
0 & 1 & 5 \
0 & 0 & 1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
1 & 0 & - 7 \
0 & 1 & - 6 \
0 & 0 & 0
\end{array} \right]$ 
E)  $\left[ \begin{array} { r r r } 
1 & 0 & - 6 \
0 & 1 & 7 \
0 & 0 & 0
\end{array} \right]$

Question 8

Use the inverse formula $A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } 
d & - b \
- c & a
\end{array} \right]$ to find the inverse of the 2×2 matrix (if it exists).   $\left[ \begin{array} { c c } 
6 & 7 \
- 5 & 13
\end{array} \right]$

Accepted Answer

A)   $\frac { 1 } { 113 } \left[ \begin{array} { c c } 
13 & - 7 \
5 & 6
\end{array} \right]$ 
B)   $\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & 7 \
5 & - 6
\end{array} \right]$ 
C)   $\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & - 7 \
5 & 6
\end{array} \right]$ 
D)   $\frac { 1 } { 113 } \left[ \begin{array} { c c } 
13 & 7 \
5 & 6
\end{array} \right]$ 
E)   $\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & - 7 \
- 5 & - 6
\end{array} \right]$ 
A)   $\frac { 1 } { 113 } \left[ \begin{array} { c c } 
13 & - 7 \
5 & 6
\end{array} \right]$ 
B)   $\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & 7 \
5 & - 6
\end{array} \right]$ 
C)   $\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & - 7 \
5 & 6
\end{array} \right]$ 
D)   $\frac { 1 } { 113 } \left[ \begin{array} { c c } 
13 & 7 \
5 & 6
\end{array} \right]$ 
E)   $\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & - 7 \
- 5 & - 6
\end{array} \right]$

Question 9

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

Accepted Answer

A) 27 
B)  $\frac { 4 } { 19 }$ 
C) 29 
D)  $\frac { 25 } { 2 }$ 
E)  $\frac { 6 } { 11 }$ 
A) 27 
B)  $\frac { 4 } { 19 }$ 
C) 29 
D)  $\frac { 25 } { 2 }$ 
E)  $\frac { 6 } { 11 }$

Question 10

Write the system of linear equations as a matrix equation AX = B, and use Gauss-Jordan elimination on the augmented matrix $$[ A \mid B ]$$ to solve for the matrix X. $$\left\{ \begin{aligned}
x _ { 1 } - 3 x _ { 2 } & = - 25 \
- 5 x _ { 1 } + 2 x _ { 2 } & = 34
\end{aligned} \right.$$

Accepted Answer

A)  $\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 25 \
- 4
\end{array} \right] = \left[ \begin{array} { c } 
34 \
7
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] = \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 4 \
- 25
\end{array} \right] = \left[ \begin{array} { c } 
7 \
34
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 4 \
7
\end{array} \right] = \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right] = \left[ \begin{array} { c } 
- 4 \
7
\end{array} \right]$ 
A)  $\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 25 \
- 4
\end{array} \right] = \left[ \begin{array} { c } 
34 \
7
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] = \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 4 \
- 25
\end{array} \right] = \left[ \begin{array} { c } 
7 \
34
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 4 \
7
\end{array} \right] = \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right]$ 
E)  $\left[ \begin{array} { c c } 
1 & - 3 \
- 5 & 2
\end{array} \right] \left[ \begin{array} { c } 
- 25 \
34
\end{array} \right] = \left[ \begin{array} { c } 
- 4 \
7
\end{array} \right]$

Question 11

Find the uncoded 1 × 3 row matrices for the message "TWO IF BY LAND"; then encode the message using the encoding matrix $\left[ \begin{array} { c c c } 
0 & - 1 & 1 \
1 & - 3 & 3 \
- 3 & - 1 & 0
\end{array} \right]$ .Show all your work.

Accepted Answer

The answer of Find the uncoded 1 × 3 row...

Question 12

Evaluate the determinant $\left| \begin{array} { c c } 
4 x & x \ln x \
4 & 6 + \ln x
\end{array} \right|$ in which the entries are functions.

Accepted Answer

A)  $16 \ln x$ 
B)  $4 - \ln x$ 
C)  $6 \ln x$ 
D)  $6 x$ 
E)  $24 x$ 
A)  $16 \ln x$ 
B)  $4 - \ln x$ 
C)  $6 \ln x$ 
D)  $6 x$ 
E)  $24 x$

Question 13

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.   $$\left\{ \begin{array} { r l c } 
- x + y - z & = - 20 \
2 x - y + z & = 29 \
3 x + 2 y + z & = 29
\end{array} \right.$$

Accepted Answer

A)  (-9, -3, 8) 
B)  (-9, -3, -8) 
C) (9, -3, 8) 
D)  (9, -3, -8) 
E)  (9, 3, 8) 
A)  (-9, -3, 8) 
B)  (-9, -3, -8) 
C) (9, -3, 8) 
D)  (9, -3, -8) 
E)  (9, 3, 8)

Question 14

Use a determinant to find an equation of the line passing through the points. ​
(0, 0), (3, 5)
​

Accepted Answer

A) 3x - 5y = 0 
B) 5x - 3y = 5 
C) 5x - 3y = 0 
D) 5x + 3y = 3 
E) 5x + 3y = 0 
A) 3x - 5y = 0 
B) 5x - 3y = 5 
C) 5x - 3y = 0 
D) 5x + 3y = 3 
E) 5x + 3y = 0

Question 15

If possible, find AB and state the order of the result.   $$A = \left[ \begin{array} { c c c } 
5 & 0 & 0 \
0 & 4 & 0 \
0 & 0 & - 3
\end{array} \right] , B = \left[ \begin{array} { c c c } 
5 & 0 & 0 \
0 & - 5 & 0 \
0 & 0 & 4
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c c c } 
25 & 0 & 0 \
0 & - 20 & 0 \
0 & 0 & - 12
\end{array} \right]$ Order: 3 × 3 
B)  $\left[ \begin{array} { c c c } 
23 & 0 & 0 \
0 & - 20 & 0 \
0 & 0 & - 12
\end{array} \right]$ Order: 3 × 3 
C)  $\left[ \begin{array} { c c c } 
27 & 0 & 0 \
0 & - 20 & 0 \
0 & 0 & - 12
\end{array} \right]$ Order: 3 × 3 
D)  $\left[ \begin{array} { c c c } 
21 & 0 & 0 \
0 & - 20 & 0 \
0 & 0 & - 12
\end{array} \right]$ Order: 3 × 3 
E)  $\left[ \begin{array} { c c c } 
29 & 0 & 0 \
0 & - 20 & 0 \
0 & 0 & - 12
\end{array} \right]$ Order: 3 × 3 
A)  $\left[ \begin{array} { c c c } 
25 & 0 & 0 \
0 & - 20 & 0 \
0 & 0 & - 12
\end{array} \right]$ Order: 3 × 3 
B)  $\left[ \begin{array} { c c c } 
23 & 0 & 0 \
0 & - 20 & 0 \
0 & 0 & - 12
\end{array} \right]$ Order: 3 × 3 
C)  $\left[ \begin{array} { c c c } 
27 & 0 & 0 \
0 & - 20 & 0 \
0 & 0 & - 12
\end{array} \right]$ Order: 3 × 3 
D)  $\left[ \begin{array} { c c c } 
21 & 0 & 0 \
0 & - 20 & 0 \
0 & 0 & - 12
\end{array} \right]$ Order: 3 × 3 
E)  $\left[ \begin{array} { c c c } 
29 & 0 & 0 \
0 & - 20 & 0 \
0 & 0 & - 12
\end{array} \right]$ Order: 3 × 3

Question 16

Write the system of linear equations as a matrix equation, AX = B.   $$\left\{ \begin{array} { l } 
2 x _ { 1 } + 3 x _ { 2 } = 8 \
4 x _ { 1 } + 5 x _ { 2 } = 16
\end{array} \right.$$

Accepted Answer

A)  $\left[ \begin{array} { l l } 
2 & 3 \
4 & 5
\end{array} \right] \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 }
\end{array} \right] = \left[ \begin{array} { c } 
8 \
16
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 
2 & 4 \
3 & 5
\end{array} \right] \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 }
\end{array} \right] = \left[ \begin{array} { c } 
8 \
16
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
2 & 3 \
4 & 5
\end{array} \right] \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 }
\end{array} \right] = \left[ \begin{array} { c } 
- 8 \
- 16
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
2 & 8 \
4 & 16
\end{array} \right] \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 }
\end{array} \right] = \left[ \begin{array} { l } 
3 \
5
\end{array} \right]$ 
E)  $\left[ \begin{array} { l l } 
2 & 3 \
4 & 5
\end{array} \right] \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 }
\end{array} \right] = \left[ \begin{array} { c } 
16 \
8
\end{array} \right]$ 
A)  $\left[ \begin{array} { l l } 
2 & 3 \
4 & 5
\end{array} \right] \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 }
\end{array} \right] = \left[ \begin{array} { c } 
8 \
16
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 
2 & 4 \
3 & 5
\end{array} \right] \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 }
\end{array} \right] = \left[ \begin{array} { c } 
8 \
16
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
2 & 3 \
4 & 5
\end{array} \right] \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 }
\end{array} \right] = \left[ \begin{array} { c } 
- 8 \
- 16
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c } 
2 & 8 \
4 & 16
\end{array} \right] \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 }
\end{array} \right] = \left[ \begin{array} { l } 
3 \
5
\end{array} \right]$ 
E)  $\left[ \begin{array} { l l } 
2 & 3 \
4 & 5
\end{array} \right] \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 }
\end{array} \right] = \left[ \begin{array} { c } 
16 \
8
\end{array} \right]$

Question 17

The currents in an electrical network are given by the solution of the system   $$\left\{ \begin{aligned}
I _ { 1 } - I _ { 2 } + I _ { 3 } & = 5 \
3 I _ { 1 } + 4 I _ { 2 } & = 19 \
I _ { 2 } + 3 I _ { 3 } & = 28
\end{aligned} \right.$$  
Where I1, I2 and I3 are measured in amperes.Solve the system of equations using matrices.

Accepted Answer

A) I1 = 5, I2 = 4, I3 = 8 
B) I1 = 4, I2 = 4, I3 = 8 
C) I1 = 3, I2 = 4, I3 = 8 
D) I1 = 2, I2 = 4, I3 = 8 
E) I1 = 1, I2 = 4, I3 = 8 
A) I1 = 5, I2 = 4, I3 = 8 
B) I1 = 4, I2 = 4, I3 = 8 
C) I1 = 3, I2 = 4, I3 = 8 
D) I1 = 2, I2 = 4, I3 = 8 
E) I1 = 1, I2 = 4, I3 = 8

Question 18

Find the determinant of the matrix.   $\left[ \begin{array} { c c } 
0 & 5 \
- 10 & 11
\end{array} \right]$

Accepted Answer

A)  $- 50$ 
B) 50 
C) 51 
D)  $- 51$ 
E)  $49$ 
A)  $- 50$ 
B) 50 
C) 51 
D)  $- 51$ 
E)  $49$

Question 19

The currents in an electrical network are given by the solutions of the system $$\left\{ \begin{array} { r l r } 
I _ { 1 } + I _ { 2 } - I _ { 3 } & = 0 \
2 I _ { 1 } + 3 I _ { 3 } & = 37 \
4 I _ { 2 } + I _ { 3 } & = 25
\end{array} \right.$$ where I1, I2, and I3 are measured in amperes.Solve the system of equations using matrices.

Accepted Answer

A) no solution 
B) I1 = 3, I2 = 3, I3 = 7 
C) I1 = 5, I2 = 4, I3 = 9 
D) I1 = 3, I2 = 3, I3 = 6 
E) I1 = 5, I2 = 9, I3 = 4 
A) no solution 
B) I1 = 3, I2 = 3, I3 = 7 
C) I1 = 5, I2 = 4, I3 = 9 
D) I1 = 3, I2 = 3, I3 = 6 
E) I1 = 5, I2 = 9, I3 = 4

Question 20

Solve the system using Gauss-Jordan elimination.   $$\left\{ \begin{aligned}
x + y + 2 z + t & = 62 \
x + 2 y + z + t & = 62 \
2 x + y + z + t & = 49 \
x + y + z + 2 t & = 62
\end{aligned} \right.$$

Accepted Answer

A)  $( 2 , - 15,15 , - 15 )$ 
B)  $( - 2,15,15,15 )$ 
C)  $( 2 , - 15,15,15 )$ 
D)  $( 2,15,15,15 )$ 
E)  $( 2,15 , - 15,15 )$ 
A)  $( 2 , - 15,15 , - 15 )$ 
B)  $( - 2,15,15,15 )$ 
C)  $( 2 , - 15,15,15 )$ 
D)  $( 2,15,15,15 )$ 
E)  $( 2,15 , - 15,15 )$

Find a value of y such that the triangle with the given vertices has an area of 8 square units. (5, 6), (5, 8), (-3, y)

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{aligned}7 x - 6 y & = - 40 \\- 5 x + 7 y & = 34\end{aligned} \right.$

Evaluate the expression. $\left[ \begin{array} { l l } 8 & 5 \\6 & 2\end{array} \right] + \left[ \begin{array} { l c } - 5 & - 6 \\- 8 & 2\end{array} \right] + \left[ \begin{array} { c c } 7 & - 5 \\5 & 8\end{array} \right]$

Solve the system of linear equations $\left\{ \begin{array} { l } 4 x + 4 y = 3 \\8 x - 20 y = - 9\end{array} \right.$ using an inverse matrix.

Find the inverse of the matrix $\left[ \begin{array} { c c } - 8 & 14 \\- 4 & 7\end{array} \right]$ (if it exists).

Find A + B. $A = \left[ \begin{array} { l l } 5 & - 5 \\2 & - 5\end{array} \right] , B = \left[ \begin{array} { c c } 2 & - 5 \\- 5 & 7\end{array} \right]$

Write the matrix in reduced row-echelon form. $\left[ \begin{array} { r r r } 4 & 5 & 11 \\3 & 1 & - 11 \\3 & 5 & 17\end{array} \right]$

Use the inverse formula $A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } d & - b \\- c & a\end{array} \right]$ to find the inverse of the 2×2 matrix (if it exists). $\left[ \begin{array} { c c } 6 & 7 \\- 5 & 13\end{array} \right]$

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

Write the system of linear equations as a matrix equation AX = B, and use Gauss-Jordan elimination on the augmented matrix $[ A \mid B ]$ to solve for the matrix X. $\left\{ \begin{aligned}x _ { 1 } - 3 x _ { 2 } & = - 25 \\- 5 x _ { 1 } + 2 x _ { 2 } & = 34\end{aligned} \right.$

Find the uncoded 1 × 3 row matrices for the message "TWO IF BY LAND"; then encode the message using the encoding matrix $\left[ \begin{array} { c c c } 0 & - 1 & 1 \\1 & - 3 & 3 \\- 3 & - 1 & 0\end{array} \right]$ .Show all your work.

Evaluate the determinant $\left| \begin{array} { c c } 4 x & x \ln x \\4 & 6 + \ln x\end{array} \right|$ in which the entries are functions.

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{array} { r l c } - x + y - z & = - 20 \\2 x - y + z & = 29 \\3 x + 2 y + z & = 29\end{array} \right.$

Use a determinant to find an equation of the line passing through the points. (0, 0), (3, 5)

If possible, find AB and state the order of the result. $A = \left[ \begin{array} { c c c } 5 & 0 & 0 \\0 & 4 & 0 \\0 & 0 & - 3\end{array} \right] , B = \left[ \begin{array} { c c c } 5 & 0 & 0 \\0 & - 5 & 0 \\0 & 0 & 4\end{array} \right]$

Write the system of linear equations as a matrix equation, AX = B. $\left\{ \begin{array} { l } 2 x _ { 1 } + 3 x _ { 2 } = 8 \\4 x _ { 1 } + 5 x _ { 2 } = 16\end{array} \right.$

Find the determinant of the matrix. $\left[ \begin{array} { c c } 0 & 5 \\- 10 & 11\end{array} \right]$

Solve the system using Gauss-Jordan elimination. $\left\{ \begin{aligned}x + y + 2 z + t & = 62 \\x + 2 y + z + t & = 62 \\2 x + y + z + t & = 49 \\x + y + z + 2 t & = 62\end{aligned} \right.$

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 a value of y such that the triangle with the given vertices has an area of 8 square units. ​ ​(5, 6), ​(5, 8), ​(-3, y) ​

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. {7x−6y=−40−5x+7y=34\left\{ \begin{aligned}7 x - 6 y & = - 40 \\- 5 x + 7 y & = 34\end{aligned} \right.{7x−6y−5x+7y​=−40=34​

Solve the system of linear equations {4x+4y=38x−20y=−9\left\{ \begin{array} { l } 4 x + 4 y = 3 \\8 x - 20 y = - 9\end{array} \right.{4x+4y=38x−20y=−9​ using an inverse matrix.

Find the inverse of the matrix [−814−47]\left[ \begin{array} { c c } - 8 & 14 \\- 4 & 7\end{array} \right][−8−4​147​] (if it exists).

Find A + B. A=[5−52−5],B=[2−5−57]A = \left[ \begin{array} { l l } 5 & - 5 \\2 & - 5\end{array} \right] , B = \left[ \begin{array} { c c } 2 & - 5 \\- 5 & 7\end{array} \right]A=[52​−5−5​],B=[2−5​−57​]

Write the matrix in reduced row-echelon form. [451131−113517]\left[ \begin{array} { r r r } 4 & 5 & 11 \\3 & 1 & - 11 \\3 & 5 & 17\end{array} \right]​433​515​11−1117​​

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

Find the uncoded 1 × 3 row matrices for the message "TWO IF BY LAND"; then encode the message using the encoding matrix [0−111−33−3−10]\left[ \begin{array} { c c c } 0 & - 1 & 1 \\1 & - 3 & 3 \\- 3 & - 1 & 0\end{array} \right]​01−3​−1−3−1​130​​ .Show all your work.

Evaluate the determinant ∣4xxln⁡x46+ln⁡x∣\left| \begin{array} { c c } 4 x & x \ln x \\4 & 6 + \ln x\end{array} \right|​4x4​xlnx6+lnx​​ in which the entries are functions.

Use a determinant to find an equation of the line passing through the points. ​ (0, 0), (3, 5) ​

Write the system of linear equations as a matrix equation, AX = B. {2x1+3x2=84x1+5x2=16\left\{ \begin{array} { l } 2 x _ { 1 } + 3 x _ { 2 } = 8 \\4 x _ { 1 } + 5 x _ { 2 } = 16\end{array} \right.{2x1​+3x2​=84x1​+5x2​=16​

Find the determinant of the matrix. [05−1011]\left[ \begin{array} { c c } 0 & 5 \\- 10 & 11\end{array} \right][0−10​511​]

Solve the system using Gauss-Jordan elimination. {x+y+2z+t=62x+2y+z+t=622x+y+z+t=49x+y+z+2t=62\left\{ \begin{aligned}x + y + 2 z + t & = 62 \\x + 2 y + z + t & = 62 \\2 x + y + z + t & = 49 \\x + y + z + 2 t & = 62\end{aligned} \right.⎩⎨⎧​x+y+2z+tx+2y+z+t2x+y+z+tx+y+z+2t​=62=62=49=62​

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 a value of y such that the triangle with the given vertices has an area of 8 square units. (5, 6), (5, 8), (-3, y)

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{aligned}7 x - 6 y & = - 40 \\- 5 x + 7 y & = 34\end{aligned} \right.$

Solve the system of linear equations $\left\{ \begin{array} { l } 4 x + 4 y = 3 \\8 x - 20 y = - 9\end{array} \right.$ using an inverse matrix.

Find the inverse of the matrix $\left[ \begin{array} { c c } - 8 & 14 \\- 4 & 7\end{array} \right]$ (if it exists).

Find A + B. $A = \left[ \begin{array} { l l } 5 & - 5 \\2 & - 5\end{array} \right] , B = \left[ \begin{array} { c c } 2 & - 5 \\- 5 & 7\end{array} \right]$

Write the matrix in reduced row-echelon form. $\left[ \begin{array} { r r r } 4 & 5 & 11 \\3 & 1 & - 11 \\3 & 5 & 17\end{array} \right]$

Find the uncoded 1 × 3 row matrices for the message "TWO IF BY LAND"; then encode the message using the encoding matrix $\left[ \begin{array} { c c c } 0 & - 1 & 1 \\1 & - 3 & 3 \\- 3 & - 1 & 0\end{array} \right]$ .Show all your work.

Evaluate the determinant $\left| \begin{array} { c c } 4 x & x \ln x \\4 & 6 + \ln x\end{array} \right|$ in which the entries are functions.

Use a determinant to find an equation of the line passing through the points. (0, 0), (3, 5)

Write the system of linear equations as a matrix equation, AX = B. $\left\{ \begin{array} { l } 2 x _ { 1 } + 3 x _ { 2 } = 8 \\4 x _ { 1 } + 5 x _ { 2 } = 16\end{array} \right.$

Find the determinant of the matrix. $\left[ \begin{array} { c c } 0 & 5 \\- 10 & 11\end{array} \right]$

Solve the system using Gauss-Jordan elimination. $\left\{ \begin{aligned}x + y + 2 z + t & = 62 \\x + 2 y + z + t & = 62 \\2 x + y + z + t & = 49 \\x + y + z + 2 t & = 62\end{aligned} \right.$