Question 1

$\left\{ \begin{array} { l } 
x + y + z = 6 \
x - z = - 2 \
y + 3 z = 11
\end{array} \right.$

Accepted Answer

A)   x=-1, y=2, z=-3 
B)   x=1, y=2, z=3 
C)   x=3, y=-2, z=5 
D)   x=0, y=4, z=2 
A)   x=-1, y=2, z=-3 
B)   x=1, y=2, z=3 
C)   x=3, y=-2, z=5 
D)   x=0, y=4, z=2

Question 2

We have encoded a message by assigning the numbers 1 - 26 to the letters a - z of the alphabet, respectively, and
assigning 27 to a blank space. We have further encoded it by using an encoding matrix. Decode this message by finding
the inverse of the encoding matrix and multiplying it times the coded message.
-The encoding matrix i $$A = \left[ \begin{array} { r r r } 
1 & 0 & 0 \
- 1 & 1 & 0 \
1 & 1 & 1
\end{array} \right] \text { and the encoded message is } 9,11,56,9,10,55,12 , - 3,43,9,5,30$$

Accepted Answer

The answer of We have encoded a message by assigning...

Question 3

The following system does not have a unique solution. Solve the system.
-$$\left\{ \begin{array} { c } 
- 2 x + 3 y + z = - 24 \
3 x - y + 5 z = 8 \
x + 2 y + 6 z = - 16 \
\end{array} \right.$$

Accepted Answer

A)  x = 0 , y = - 8 , z = 0 
B)  infinitely many solutions of the form $x = - \frac { 16 } { 7 } z , y = - 8 - \frac { 13 } { 7 } z , z = z$ 
C)  inconsistent system, no solution 
D)  $x = - \frac { 16 } { 7 } , y = \frac { 43 } { 7 } , z = 1$ 
A)  x = 0 , y = - 8 , z = 0 
B)  infinitely many solutions of the form $x = - \frac { 16 } { 7 } z , y = - 8 - \frac { 13 } { 7 } z , z = z$ 
C)  inconsistent system, no solution 
D)  $x = - \frac { 16 } { 7 } , y = \frac { 43 } { 7 } , z = 1$

Question 4

Solve using Cramer's Rule.
-Two different gasohol mixtures are available. One contains 5% alcohol and the other 12% alcohol. How much of each should be mixed to obtain 1000 gallons of gasohol containing 10% alcohol?

Accepted Answer

A)  714 gal 5%, 714 gal 12% 
B)  50 gal 5%, 120 gal 12% 
C)  417 gal 5%, 583 gal 12% 
D)  286 gal 5%, 714 gal 12% 
A)  714 gal 5%, 714 gal 12% 
B)  50 gal 5%, 120 gal 12% 
C)  417 gal 5%, 583 gal 12% 
D)  286 gal 5%, 714 gal 12%

Question 5

Find the indicated sum or difference, if it is defined.
-$$\left[ \begin{array} { l l } 
- 1 & - 6 \
- 1 & - 8 \
- 6 & - 5
\end{array} \right] + \left[ \begin{array} { r r } 
6 & 3 \
- 2 & - 2 \
- 5 & - 6
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { r r } 5 & - 3 \ - 3 & - 10 \ - 11 & - 11 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 7 & - 9 \ 1 & - 6 \ - 1 & - 12 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 5 & - 8 \ - 3 & - 10 \ - 11 & - 11 \end{array} \right]$ 
D)  not defined 
A)  $\left[ \begin{array} { r r } 5 & - 3 \ - 3 & - 10 \ - 11 & - 11 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 7 & - 9 \ 1 & - 6 \ - 1 & - 12 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 5 & - 8 \ - 3 & - 10 \ - 11 & - 11 \end{array} \right]$ 
D)  not defined

Question 6

Decide whether or not matrix A and matrix B are inverses.
-$$A = \left[ \begin{array} { r r } 
6 & - 5 \
- 3 & 5
\end{array} \right] \text { and } B = \left[ \begin{array} { l l } 
1 / 3 & 1 / 3 \
1 / 5 & 2 / 5
\end{array} \right]$$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 7

first two games of the season. Write a matrix containing the total number of points and rebounds for each of the starting
five. 
$$\begin{array}{l}
\begin{array} { l | c | c } 
\text { Game } 1 & \text { Points } & \text { Rebounds } \
\hline \text { Levy } & 20 & 3 \
\text { Cowens } & 16 & 5 \
\text { Williams } & 8 & 12 \
\text { Miller } & 6 & 11 \
\text { Jenkins } & 10 & 2
\end{array}\\
\begin{array} { l | c | c } 
\text { Game } 2 & \text { Points } & \text { Rebounds } \
\hline \text { Levy } & 18 & 4 \
\text { Cowens } & 14 & 3 \
\text { Williams } & 12 & 9 \
\text { Miller } & 4 & 10 \
\text { Jenkins } & 10 & 3
\end{array}
\end{array}$$

Accepted Answer

A)  $\left[ \begin{array} { l l } 5 & 62 \end{array} \right]$
В) $\left[ \begin{array} { r r } 7 & 38 \ 30 & 8 \ 21 & 20 \ 21 & 10 \ 5 & 20 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 38 & 7 \ 30 & 8 \ 20 & 21 \ 10 & 21 \ 20 & 5 \end{array} \right]$ 
D)  $[ 625 ]$ Answer: C
-A bakery sells four main items: rolls, bread, cake, and pie. The amount of each of five ingredients (in cups, except for eggs) required to make a dozen rolls, a loaf of bread, a cake, or a pie is given by matrix A.  
Suppose a day's orders total 20 dozen rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a $1 \times 4$ matrix and, using matrix multiplication, find a matrix for the amount of each ingredient needed to fill the day's orders. 
A)  $\left[ \begin{array} { l l l l } 910 & 952 & 135 & 125 \end{array} \right)$ 
B)  These matrices cannot be multiplied. 
C)  $\left[ \begin{array} { l l l } 730 & 700150125 & 250 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l l l } 910 & 1060 & 105 & 70 \end{array} \right]$ 
A)  $\left[ \begin{array} { l l } 5 & 62 \end{array} \right]$
В) $\left[ \begin{array} { r r } 7 & 38 \ 30 & 8 \ 21 & 20 \ 21 & 10 \ 5 & 20 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 38 & 7 \ 30 & 8 \ 20 & 21 \ 10 & 21 \ 20 & 5 \end{array} \right]$ 
D)  $[ 625 ]$ Answer: C
-A bakery sells four main items: rolls, bread, cake, and pie. The amount of each of five ingredients (in cups, except for eggs) required to make a dozen rolls, a loaf of bread, a cake, or a pie is given by matrix A.  
Suppose a day's orders total 20 dozen rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a $1 \times 4$ matrix and, using matrix multiplication, find a matrix for the amount of each ingredient needed to fill the day's orders. 
A)  $\left[ \begin{array} { l l l l } 910 & 952 & 135 & 125 \end{array} \right)$ 
B)  These matrices cannot be multiplied. 
C)  $\left[ \begin{array} { l l l } 730 & 700150125 & 250 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l l l } 910 & 1060 & 105 & 70 \end{array} \right]$

Question 8

$\left| \begin{array} { r r r } 
0 & 0 & 2 \
1 & - 9 & 5 \
0 & 7 & 6
\end{array} \right|$

Accepted Answer

A)  14 
B)   -40 
C)   -68 
D)   -14 
A)  14 
B)   -40 
C)   -68 
D)   -14

Question 9

Find the value of the determinant.
-$$\left| \begin{array} { r r } 
\frac { 1 } { 7 } & - \frac { 1 } { 6 } \
6 & 7
\end{array} \right|$$

Accepted Answer

A)  0 
B)  $- 2$ 
C)  2 
D)  $\frac { 85 } { 42 }$ 
A)  0 
B)  $- 2$ 
C)  2 
D)  $\frac { 85 } { 42 }$

Question 10

Find the inverse matrix of A.
-$$A = \left[ \begin{array} { l l l } 
1 & 1 & 1 \
2 & 1 & 1 \
2 & 2 & 3
\end{array} \right]$$

Accepted Answer

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

Question 11

Solve for x.
-$$\left| \begin{array} { l l } 
x - 6 & - 12 \
x - 5 & - 10
\end{array} \right| = 22$$

Accepted Answer

A)  11 
B)  $- 132$ 
C)  13 
D)  0 
A)  11 
B)  $- 132$ 
C)  13 
D)  0

Question 12

Compute AB, if possible.
-$A = \left[ \begin{array} { r r } 3 & - 2 \ 3 & 0 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 0 & - 2 \ 3 & 6 \end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { r r } - 6 & - 18 \ 0 & - 6 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 18 & - 6 \ - 6 & 0 \end{array} \right]$ 
C)  not possible 
D)  $\left[ \begin{array} { r r } - 6 & 0 \ 27 & - 6 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r } - 6 & - 18 \ 0 & - 6 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 18 & - 6 \ - 6 & 0 \end{array} \right]$ 
C)  not possible 
D)  $\left[ \begin{array} { r r } - 6 & 0 \ 27 & - 6 \end{array} \right]$

Question 13

Use an inverse matrix to find the solution to the system.
-$$\left\{ \begin{array} { c } 
- 4 x + 3 y - z = - 3 \
x - 6 y + 8 z = - 5 \
8 x + y + z = 51
\end{array} \right.$$

Accepted Answer

A)  $x = 5 , y = 7 , z = 4$ 
B)  $x = - 5 , y = 7 , z = 10$ 
C)  $x = 5 , y = 4 , z = 7$ 
D)  $x = - 5 , y = 10 , z = 7$ 
A)  $x = 5 , y = 7 , z = 4$ 
B)  $x = - 5 , y = 7 , z = 10$ 
C)  $x = 5 , y = 4 , z = 7$ 
D)  $x = - 5 , y = 10 , z = 7$

Question 14

We can encode a message by assigning the numbers 1 - 26 to the letters a - z of the alphabet, respectively, and assigning
27 to a blank space. To further encode a message, we can use an encoding matrix A to convert these numbers into new
pairs or triples of numbers. Use matrix A to encode the given message.
-Convert "I love you" from letters to numbers. Use pairs of those numbers, in order, to create $2 \times 1$ matrices, and multiply $\mathrm { A } = \left[ \begin{array} { l l } 2 & 1 \ 1 & 3 \end{array} \right]$ times those matrices to encode the message "I love you" into pairs of coded numbers.
Combine the pairs of numbers to give the encoded numerical message.

Accepted Answer

45, 90, 39

Question 15

Decide whether or not matrix A and matrix B are inverses.
-$$A = \left[ \begin{array} { r r r } 
1 & 0 & 0 \
- 1 & 1 & 0 \
1 & 1 & 1
\end{array} \right] \text { and } B = \left[ \begin{array} { r r r } 
1 & 0 & 0 \
1 & 1 & 0 \
- 2 & - 1 & 1
\end{array} \right]$$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 16

first two games of the season. Write a matrix containing the total number of points and rebounds for each of the starting
five. 
$$\begin{array}{l}
\begin{array} { l | c | c } 
\text { Game } 1 & \text { Points } & \text { Rebounds } \
\hline \text { Levy } & 20 & 3 \
\text { Cowens } & 16 & 5 \
\text { Williams } & 8 & 12 \
\text { Miller } & 6 & 11 \
\text { Jenkins } & 10 & 2
\end{array}\\
\begin{array} { l | c | c } 
\text { Game } 2 & \text { Points } & \text { Rebounds } \
\hline \text { Levy } & 18 & 4 \
\text { Cowens } & 14 & 3 \
\text { Williams } & 12 & 9 \
\text { Miller } & 4 & 10 \
\text { Jenkins } & 10 & 3
\end{array}
\end{array}$$

Accepted Answer

A)  $\left[ \begin{array} { l l } 5 & 62 \end{array} \right]$
В) $\left[ \begin{array} { r r } 7 & 38 \ 30 & 8 \ 21 & 20 \ 21 & 10 \ 5 & 20 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 38 & 7 \ 30 & 8 \ 20 & 21 \ 10 & 21 \ 20 & 5 \end{array} \right]$ 
D)  $[ 625 ]$ Answer: C
-The tables below show the times by three runners in two heats of a race. Write a matrix that shows the increase (decrease) in each time from the first heat to the second. $\begin{array}{l|l}
\text { Heat 1 } & \begin{array}{l}
\text { Time } \
\text { (sec.) }
\end{array} \
\hline \text { Russell } & 15.5 \
\text { Sergy } & 15.8 \
\text { Omar } & 162
\end{array}$

$\begin{array}{l|l}
\text { Heat } 2 & \begin{array}{l}
\text { Time } \
(\mathrm{sec})
\end{array} \
\hline \text { Russell } & 15.3 \
\text { Sergy } & 15.4 \
\text { Omar } & 15.7
\end{array}$ 
A)  $\left[ \begin{array} { l l } 150 & 15.8 \ 15 & 16.2 \ 15.5 & 15.3 \end{array} \right]$ 
B)  $\left[ \begin{array} { l } - 0.2 \ - 0.4 \ - 0.3 \end{array} \right]$ 
C)  $\left[ \begin{array} { r } - 0.2 \ - 0.4 \ 0.5 \end{array} \right]$ 
D)  $\left[ \begin{array} { l } - 0.2 \ - 0.4 \ - 0.5 \end{array} \right]$ 
A)  $\left[ \begin{array} { l l } 5 & 62 \end{array} \right]$
В) $\left[ \begin{array} { r r } 7 & 38 \ 30 & 8 \ 21 & 20 \ 21 & 10 \ 5 & 20 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 38 & 7 \ 30 & 8 \ 20 & 21 \ 10 & 21 \ 20 & 5 \end{array} \right]$ 
D)  $[ 625 ]$ Answer: C
-The tables below show the times by three runners in two heats of a race. Write a matrix that shows the increase (decrease) in each time from the first heat to the second. $\begin{array}{l|l}
\text { Heat 1 } & \begin{array}{l}
\text { Time } \
\text { (sec.) }
\end{array} \
\hline \text { Russell } & 15.5 \
\text { Sergy } & 15.8 \
\text { Omar } & 162
\end{array}$

$\begin{array}{l|l}
\text { Heat } 2 & \begin{array}{l}
\text { Time } \
(\mathrm{sec})
\end{array} \
\hline \text { Russell } & 15.3 \
\text { Sergy } & 15.4 \
\text { Omar } & 15.7
\end{array}$ 
A)  $\left[ \begin{array} { l l } 150 & 15.8 \ 15 & 16.2 \ 15.5 & 15.3 \end{array} \right]$ 
B)  $\left[ \begin{array} { l } - 0.2 \ - 0.4 \ - 0.3 \end{array} \right]$ 
C)  $\left[ \begin{array} { r } - 0.2 \ - 0.4 \ 0.5 \end{array} \right]$ 
D)  $\left[ \begin{array} { l } - 0.2 \ - 0.4 \ - 0.5 \end{array} \right]$

Question 17

$\left\{ \begin{aligned}
6 y + w & = - 25 \
x + y + 4 z - w & = 14 \
6 x + z + 4 w & = 6 \
x + y + 4 z & = 13
\end{aligned} \right.$

Accepted Answer

A)   x=1, y=-4, z=4, w=-1 
B)   x=6, y=-24, z=16, w=-6 
C)   x=1, y=-24, z=8, w=-2 
D)   x=1, y=-4, z=16, w=-1 
A)   x=1, y=-4, z=4, w=-1 
B)   x=6, y=-24, z=16, w=-6 
C)   x=1, y=-24, z=8, w=-2 
D)   x=1, y=-4, z=16, w=-1

Question 18

$\left\{ \begin{array} { l } 
\sqrt { x } - y = 0 \
x ^ { 2 } + 2 y = 20
\end{array} \right.$

Accepted Answer

A)   (4,2) 
B)   (0,10) 
C)   (2,4) 
D)   (4,2),(-4,2) 
A)   (4,2) 
B)   (0,10) 
C)   (2,4) 
D)   (4,2),(-4,2)

Question 19

Compute AB, if possible.
-$A = \left[ \begin{array} { l l l } - 8 & 2 & 4 \end{array} \right]$ and $B = \left[ \begin{array} { r } 5 \ 0 \ - 3 \end{array} \right]$

Accepted Answer

A)  $[ - 52 ]$ 
B)  $[ 154 ]$ 
C)  $\left[ \begin{array} { r } - 40 \ 0 \ - 12 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l l } - 40 & 0 & - 12 \end{array} \right]$ 
A)  $[ - 52 ]$ 
B)  $[ 154 ]$ 
C)  $\left[ \begin{array} { r } - 40 \ 0 \ - 12 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l l } - 40 & 0 & - 12 \end{array} \right]$

Question 20

We have encoded a message by assigning the numbers 1 - 26 to the letters a - z of the alphabet, respectively, and
assigning 27 to a blank space. We have further encoded it by using an encoding matrix. Decode this message by finding
the inverse of the encoding matrix and multiplying it times the coded message.
-The encoding matrix i $$A = \left[ \begin{array} { l l } 
1 & 1 \
3 & 2
\end{array} \right] \text { and the encoded message is } 29,63,28,72,10,21,41,96,29,67$$

Accepted Answer

The answer of We have encoded a message by assigning...

$\left\{ \begin{array} { l } x + y + z = 6 \\x - z = - 2 \\y + 3 z = 11\end{array} \right.$

The following system does not have a unique solution. Solve the system. - $\left\{ \begin{array} { c } - 2 x + 3 y + z = - 24 \\3 x - y + 5 z = 8 \\x + 2 y + 6 z = - 16 \\\end{array} \right.$

Solve using Cramer's Rule. -Two different gasohol mixtures are available. One contains 5% alcohol and the other 12% alcohol. How much of each should be mixed to obtain 1000 gallons of gasohol containing 10% alcohol?

Find the indicated sum or difference, if it is defined. - $\left[ \begin{array} { l l } - 1 & - 6 \\- 1 & - 8 \\- 6 & - 5\end{array} \right] + \left[ \begin{array} { r r } 6 & 3 \\- 2 & - 2 \\- 5 & - 6\end{array} \right]$

Decide whether or not matrix A and matrix B are inverses. - $A = \left[ \begin{array} { r r } 6 & - 5 \\- 3 & 5\end{array} \right] \text { and } B = \left[ \begin{array} { l l } 1 / 3 & 1 / 3 \\1 / 5 & 2 / 5\end{array} \right]$

$\left| \begin{array} { r r r } 0 & 0 & 2 \\1 & - 9 & 5 \\0 & 7 & 6\end{array} \right|$

Find the value of the determinant. - $\left| \begin{array} { r r } \frac { 1 } { 7 } & - \frac { 1 } { 6 } \\6 & 7\end{array} \right|$

Find the inverse matrix of A. - $A = \left[ \begin{array} { l l l } 1 & 1 & 1 \\2 & 1 & 1 \\2 & 2 & 3\end{array} \right]$

Solve for x. - $\left| \begin{array} { l l } x - 6 & - 12 \\x - 5 & - 10\end{array} \right| = 22$

Compute AB, if possible. - $A = \left[ \begin{array} { r r } 3 & - 2 \\ 3 & 0 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 0 & - 2 \\ 3 & 6 \end{array} \right]$

Use an inverse matrix to find the solution to the system. - $\left\{ \begin{array} { c } - 4 x + 3 y - z = - 3 \\x - 6 y + 8 z = - 5 \\8 x + y + z = 51\end{array} \right.$

Decide whether or not matrix A and matrix B are inverses. - $A = \left[ \begin{array} { r r r } 1 & 0 & 0 \\- 1 & 1 & 0 \\1 & 1 & 1\end{array} \right] \text { and } B = \left[ \begin{array} { r r r } 1 & 0 & 0 \\1 & 1 & 0 \\- 2 & - 1 & 1\end{array} \right]$

$\left\{ \begin{aligned}6 y + w & = - 25 \\x + y + 4 z - w & = 14 \\6 x + z + 4 w & = 6 \\x + y + 4 z & = 13\end{aligned} \right.$

$\left\{ \begin{array} { l } \sqrt { x } - y = 0 \\x ^ { 2 } + 2 y = 20\end{array} \right.$

Compute AB, if possible. - $A = \left[ \begin{array} { l l l } - 8 & 2 & 4 \end{array} \right]$ and $B = \left[ \begin{array} { r } 5 \\ 0 \\ - 3 \end{array} \right]$

Functions, Graphs, and Models; Linear Functions

Linear Models, Equations, and Inequalities

Quadratic, Piecewise-Defined, and Power Functions

Additional Topics With Functions

Exponential and Logarithmic Functions

Higher-Degree Polynomial and Rational Functions

Special Topics in Algebra

Filters

Exam 7: Systems of Equations and Matrices

{x+y+z=6x−z=−2y+3z=11\left\{ \begin{array} { l } x + y + z = 6 \\x - z = - 2 \\y + 3 z = 11\end{array} \right.⎩⎨⎧​x+y+z=6x−z=−2y+3z=11​

The following system does not have a unique solution. Solve the system. - {−2x+3y+z=−243x−y+5z=8x+2y+6z=−16\left\{ \begin{array} { c } - 2 x + 3 y + z = - 24 \\3 x - y + 5 z = 8 \\x + 2 y + 6 z = - 16 \\\end{array} \right.⎩⎨⎧​−2x+3y+z=−243x−y+5z=8x+2y+6z=−16​

Solve using Cramer's Rule. -Two different gasohol mixtures are available. One contains 5% alcohol and the other 12% alcohol. How much of each should be mixed to obtain 1000 gallons of gasohol containing 10% alcohol?

∣0021−95076∣\left| \begin{array} { r r r } 0 & 0 & 2 \\1 & - 9 & 5 \\0 & 7 & 6\end{array} \right|​010​0−97​256​​

Find the value of the determinant. - ∣17−1667∣\left| \begin{array} { r r } \frac { 1 } { 7 } & - \frac { 1 } { 6 } \\6 & 7\end{array} \right|​71​6​−61​7​​

Find the inverse matrix of A. - A=[111211223]A = \left[ \begin{array} { l l l } 1 & 1 & 1 \\2 & 1 & 1 \\2 & 2 & 3\end{array} \right]A=​122​112​113​​

Solve for x. - ∣x−6−12x−5−10∣=22\left| \begin{array} { l l } x - 6 & - 12 \\x - 5 & - 10\end{array} \right| = 22​x−6x−5​−12−10​​=22

Compute AB, if possible. - A=[3−230]A = \left[ \begin{array} { r r } 3 & - 2 \\ 3 & 0 \end{array} \right]A=[33​−20​] and B=[0−236]B = \left[ \begin{array} { r r } 0 & - 2 \\ 3 & 6 \end{array} \right]B=[03​−26​]

Use an inverse matrix to find the solution to the system. - {−4x+3y−z=−3x−6y+8z=−58x+y+z=51\left\{ \begin{array} { c } - 4 x + 3 y - z = - 3 \\x - 6 y + 8 z = - 5 \\8 x + y + z = 51\end{array} \right.⎩⎨⎧​−4x+3y−z=−3x−6y+8z=−58x+y+z=51​

{6y+w=−25x+y+4z−w=146x+z+4w=6x+y+4z=13\left\{ \begin{aligned}6 y + w & = - 25 \\x + y + 4 z - w & = 14 \\6 x + z + 4 w & = 6 \\x + y + 4 z & = 13\end{aligned} \right.⎩⎨⎧​6y+wx+y+4z−w6x+z+4wx+y+4z​=−25=14=6=13​

{x−y=0x2+2y=20\left\{ \begin{array} { l } \sqrt { x } - y = 0 \\x ^ { 2 } + 2 y = 20\end{array} \right.{x​−y=0x2+2y=20​

Compute AB, if possible. - A=[−824]A = \left[ \begin{array} { l l l } - 8 & 2 & 4 \end{array} \right]A=[−8​2​4​] and B=[50−3]B = \left[ \begin{array} { r } 5 \\ 0 \\ - 3 \end{array} \right]B=​50−3​​