Deck 6: Matrices and Determinants

Question 1

Use Matrices and Gaussian Elimination to Solve Systems
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
$$\begin{aligned}x + y + z - w & = 6 \\2 x - y + 3 z + 4 w & = - 4 \\4 x + 2 y - z - w & = - 13 \\- x - 2 y + 4 z + 3 w = & 12\end{aligned}$$

A) $\{ ( - 4,3,5 , - 2 ) \}$
B) $\{ ( 4 , - 3 , - 5,2 ) \}$
C) $\left\{ \left( - \frac { 1 } { 4 } , \frac { 1 } { 3 } , \frac { 1 } { 5 } , - \frac { 1 } { 2 } \right) \right\}$
D) $\left\{ \left( \frac { 1 } { 4 } , - \frac { 1 } { 3 } , - \frac { 1 } { 5 } , \frac { 1 } { 2 } \right) \right\}$

Answer 1

A

Question 2

Use Matrices and Gauss-Jordan Elimination to Solve Systems
Solve the system of equations using matrices. Use Gauss-Jordan elimination.
$$\begin{array} { l } x = 6 - y - z \\x - y + 3 z = - 6 \\2 x + y = 8 - z\end{array}$$

A) $\{ ( 2,5 , - 1 ) \}$
B) $\{ ( 5 , - 1,2 ) \}$
C) $\{ ( - 1,5,2 ) \}$
D) $\{ ( - 1,2,5 ) \}$

Answer 2

A

Question 3

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables. Then use back-substitution to find the solution.
$\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & - 6 \\ 0 & 1 & - 4 & 8 & 0 \\ 0 & 0 & 1 & 4 & 13 \\ 0 & 0 & 0 & 1 & - 3 \end{array} \right]$

A) $\{ ( - 102,124,25 , - 3 ) \}$
B) $\{ ( - 6,0,13 , - 3 ) \}$
C) $\{ ( - 8 , - 5,8 , - 4 ) \}$
D) $\{ ( - 3,25,124 , - 102 ) \}$

Answer 3

A

Question 4

Use Matrices and Gaussian Elimination to Solve Systems
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
$$\begin{aligned}x - y + 4 z & = 20 \\5 x + z & = 4 \\x + 3 y + z & = - 8\end{aligned}$$

A) $\{ ( 0 , - 4,4 ) \}$
B) $\{ ( 0,4 , - 4 ) \}$
C) $\{ ( 4 , - 4,0 ) \}$
D) $\{ ( 4,0 , - 4 ) \}$

Question 5

Use Matrices and Gauss-Jordan Elimination to Solve Systems
Solve the system of equations using matrices. Use Gauss-Jordan elimination.
$$\begin{array} { r r } 6 x - 7 y - z = & 9 \\x + 4 y - 7 z = & - 12 \\- 2 x + y + z = & - 5\end{array}$$

A) $\{ ( 7,4,5 ) \}$
B) $\{ ( 7,5,4 ) \}$
C) $\{ ( - 7,4,14 ) \}$
D) $\{ ( 14,4 , - 7 ) \}$

Question 6

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables.
$$\left[ \begin{array} { r r r r | r } 6 & 1 & 0 & 8 & - 6 \\- 1 & 3 & 1 & 0 & 7 \\7 & 0 & 0 & 5 & - 4 \\0 & 4 & 0 & - 4 & 10\end{array} \right]$$

A)
$$\begin{array} { r } 6 x + y + 8 w = - 6 \\- x + 3 y + z = 7 \\7 x + 5 w = - 4 \\4 y - 4 w = 10\end{array}$$
B)
$6 x + y + z + 8 w = - 6$
$- x + 3 y + z + w = 7$
$7 x + y + z + 5 w = - 4$
C)
$\begin{array}{r}6 x+y+8 w=-6 \\x+3 y+z=7 \\7 x+5 w=-4 \\4 y+4 w=10\end{array}$


D)
$\begin{array}{r}6 x+y+8 z=-6 \\-x+3 y+z=7 \\7 x+5 y=-4 \\4 x-4 y=10\end{array}$

Question 7

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables.
$$\begin{array} { r } 7 x + 8 y - 12 z + w = 9 \\4 y + z = 10 \\x - y - 3 z = - 1 \\5 x - 5 y + 2 z = 2\end{array}$$

A)
$$\left[ \begin{array} { r r r r | r } 7 & 8 & - 12 & 1 & 9 \\0 & 4 & 1 & 0 & 10 \\1 & - 1 & - 3 & 0 & - 1 \\5 & - 5 & 2 & 0 & 2\end{array} \right]$$
B)
$$\left[ \begin{array} { r r r | r } 7 & 8 & - 12 & 9 \\0 & 4 & 1 & 6 \\1 & - 1 & - 3 & - 1 \\5 & - 5 & 2 & 2\end{array} \right]$$
C)
$$\left[ \begin{array} { l r r | r } 7 & 0 & 1 & 9 \\8 & 4 & - 1 & - 5 \\- 12 & 1 & - 3 & 2 \\1 & 0 & 0 & 0 \\9 & 10 & - 1 & 2\end{array} \right]$$
D)
$$\left[ \begin{array} { r r r r | r } 7 & 8 & 12 & 1 & 9 \\0 & 4 & 1 & 0 & 10 \\1 & 1 & 3 & 0 & - 1 \\5 & 5 & 2 & 0 & 2\end{array} \right]$$

Question 8

Use Matrices and Gauss-Jordan Elimination to Solve Systems
Solve the system of equations using matrices. Use Gauss-Jordan elimination.
$$\begin{array} { r } - 4 x - y - 3 z = - 39 \\7 x + 8 y - 4 z = 35 \\3 x - 2 y + z = - 4\end{array}$$

A) $\{ ( 1,8,9 ) \}$
B) $\{ ( 1,9,8 ) \}$
C) $\{ ( - 1,8,2 ) \}$
D) $\{ ( 2,8 , - 1 ) \}$

Question 9

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables. Then use back-substitution to find the solution.
$\left[ \begin{array} { r r r | r } 1 & 6 & 7 & - 3 \\ 0 & 1 & 9 & 3 \\ 0 & 0 & 1 & - 8 \end{array} \right]$

A) $\{ ( - 397,75 , - 8 ) \}$
В) $\{ ( - 3,3 , - 8 ) \}$
C) $\{ ( - 17 , - 7 , - 9 ) \}$
D) $\{ ( - 473 , - 69 , - 8 ) \}$

Question 10

Use Matrices and Gaussian Elimination to Solve Systems
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
$$\begin{aligned}- 5 x - y + 6 z & = - 29 \\- 8 x - 9 z & = - 99 \\2 y + z & = 7\end{aligned}$$

A) $\{ ( 9,2,3 ) \}$
B) $\{ ( 9,3,2 ) \}$
C) $\{ ( - 9,2,18 ) \}$
D) $\{ ( - 9,18,2 ) \}$

Question 11

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables. Then use back-substitution to find the solution.
$\left[ \begin{array} { r r r | r } 1 & \frac { 5 } { 2 } & 1 & \frac { 9 } { 2 } \\ 0 & 1 & \frac { 3 } { 2 } & 8 \\ 0 & 0 & 1 & - 4 \end{array} \right]$

A) $\left\{ \left( - \frac { 53 } { 2 } , 14 , - 4 \right) \right\}$
B) $\left\{ \left( \frac { 9 } { 2 } , 8 , - 4 \right) \right\}$
C) $\left\{ \left( 0 , \frac { 11 } { 2 } , - 5 \right) \right\}$
D) $\left\{ \left( \frac { 11 } { 2 } , 14 , - 4 \right) \right\}$

Question 12

Write the augmented matrix for the system of equations.
$$\begin{array} { r } x - 5 y + z = 11 \\y + 7 z = 19 \\z = 15\end{array}$$

A)
$\left[ \begin{array} { r r r | r } 1 & - 5 & 1 & 11 \\ 0 & 1 & 7 & 19 \\ 0 & 0 & 1 & 15 \end{array} \right]$
B)
$\left[ \begin{array} { r r r | r } 0 & - 5 & 0 & 11 \\ 0 & 0 & 7 & 19 \\ 0 & 0 & 0 & 15 \end{array} \right]$
C)
$\left[ \begin{array} { l l l | l } 1 & 5 & 1 & 11 \\ 0 & 1 & 7 & 19 \\ 0 & 0 & 1 & 15 \end{array} \right]$
D)
$\left[ \begin{array} { r r r | r } 1 & - 5 & 1 & 11 \\ 1 & 1 & 7 & 19 \\ 1 & 1 & 1 & 15 \end{array} \right]$

Question 13

Use Matrices and Gaussian Elimination to Solve Systems
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
$$\begin{aligned}x + y + z & = - 5 \\x - y + 4 z & = - 13 \\4 x + y + z & = - 2\end{aligned}$$

A) $\{ ( 1 , - 2 , - 4 ) \}$
B) $\{ ( 1 , - 4 , - 2 ) \}$
C) $\{ ( - 4 , - 2,1 ) \}$
D) $\{ ( - 4,1 , - 2 ) \}$

Question 14

Perform the matrix row operation (or operations)and write the new matrix.
$\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 2 \\ 0 & - 1 & 1 & - 3 & 0 \\ 5 & 0 & - 4 & - 5 & 5 \\ - 2 & 4 & 0 & 2 & - 1 \end{array} \right] \quad \begin{array} { r } - 4 R _ { 1 } + R _ { 3 } \\ 2 R _ { 1 } + R _ { 4 } \end{array}$

A)
$\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 2 \\ 0 & - 1 & 1 & - 3 & 0 \\ 1 & - 4 & 0 & - 9 & - 3 \\ 0 & 6 & - 2 & 4 & 3 \end{array} \right]$
B)
$\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 2 \\ 0 & - 1 & 1 & - 3 & 0 \\ 1 & - 4 & 0 & - 9 & - 3 \\ - 2 & 4 & 0 & 2 & - 1 \end{array} \right]$
C)
$$\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 2 \\0 & - 1 & 1 & - 3 & 0 \\9 & 4 & - 8 & - 1 & 13 \\0 & 6 & - 2 & 4 & 3\end{array} \right]$$
D)
$$\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 2 \\0 & - 1 & 1 & - 3 & 0 \\1 & - 4 & 0 & - 9 & 1 \\0 & 6 & - 2 & 4 & 3\end{array} \right]$$

Question 15

Use Matrices and Gaussian Elimination to Solve Systems
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
$$\begin{array} { r r } 3 x + 5 y - 2 w = & - 13 \\2 x + 7 z - w = & - 1 \\4 y + 3 z + 3 w = & 1 \\- x + 2 y + 4 z = & - 5\end{array}$$

A) $\{ ( 1 , - 2,0,3 ) \}$
B) $\left\{ \left( \frac { 4 } { 3 } , - \frac { 13 } { 20 } , 0 , \frac { 5 } { 2 } \right) \right\}$
C) $\left\{ \left( \frac { 3 } { 4 } , - 2,0 , \frac { 3 } { 4 } \right) \right\}$
D) $\left\{ \left( - 1 , - \frac { 20 } { 13 } , 0 , \frac { 2 } { 5 } \right) \right\}$

Question 16

Write the augmented matrix for the system of equations.
$$\begin{array} { l } 6 x + 9 y + 6 z = 54 \\4 x + 2 y + 7 z = 17 \\3 x - 2 y + 2 z = 2\end{array}$$

A)
$$\left[ \begin{array} { r r r | r } 6 & 9 & 6 & 54 \\4 & 2 & 7 & 17 \\3 & - 2 & 2 & 2\end{array} \right]$$
B)
$\left[ \begin{array} { r r r | r } 6 & 4 & 3 & 54 \\ 9 & 2 & - 2 & 17 \\ 6 & 7 & 2 & 2 \end{array} \right]$
C)
$\left[ \begin{array} { r r r | r } 54 & 6 & 9 & 6 \\ 17 & 7 & 2 & 4 \\ 2 & 2 & - 2 & 3 \end{array} \right]$
D)
$\left[ \begin{array} { r r r } 6 & 9 & 6 \\ 4 & 2 & 7 \\ 3 & - 2 & 2 \end{array} \right]$

Question 17

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables.
$\left[ \begin{array} { r r r | r } 6 & 9 & 6 & - 2 \\ 2 & 0 & 7 & 4 \\ 6 & 8 & 0 & 2 \end{array} \right]$

A) $6 x + 9 y + 6 z = - 2$
$2 x + 7 z = 4$
$6 x + 8 y = 2$
B) $6 x - 9 y + 6 z = - 2$
$2 x + 7 z = - 4$
$6 x + 8 y = - 2$
C) $6 x + 9 y + 6 z = - 2$
$2 x + 7 z = 4$
$6 x + 8 z = 2$
D) $6 x + 9 y + 6 z = - 2$
$2 x + y + 7 z = 4$
$6 x + 8 y + z = 2$

Question 18

Write the augmented matrix for the system of equations.
$$\begin{array} { r } 4 x + 3 z = 33 \\9 y + 5 z = 42 \\6 x - 2 y + 8 z = 54\end{array}$$

A)
$\left[ \begin{array} { r r r | r } 4 & 0 & 6 & 33 \\ 0 & 9 & - 2 & 42 \\ 3 & 5 & 8 & 54 \end{array} \right]$
B)
$\left[ \begin{array} { r r r | r } 4 & 3 & 0 & 33 \\ 9 & 5 & 0 & 42 \\ 6 & - 2 & 8 & 54 \end{array} \right]$

C)
$\left[\begin{array}{rrr|r}4 & 3 & 0 & 33 \\9 & 5 & 0 & 42 \\6 & -2 & 8 & 54\end{array}\right]$

D)
$\left[ \begin{array} { r r r } 4 & 0 & 3 \\ 0 & 9 & 5 \\ 6 & - 2 & 8 \end{array} \right]$

Question 19

Perform the matrix row operation (or operations)and write the new matrix.
$\left[ \begin{array} { r r r | r } 36 & 30 & 33 & - 36 \\ 1 & 13 & - 3 & 0 \\ 2 & - 7 & 4 & 21 \end{array} \right] \frac { 1 } { 3 } \mathrm { R } _ { 1 }$

A)
$\left[ \begin{array} { c c r | r } 12 & 10 & 11 & - 12 \\ 1 & 13 & - 3 & 0 \\ 2 & - 7 & 4 & 21 \end{array} \right]$
B)
$\left[ \begin{array} { r r r | r } 36 & 30 & 33 & - 36 \\ \frac { 1 } { 3 } & \frac { 13 } { 3 } & - 1 & 0 \\ 2 & - 7 & 4 & 21 \end{array} \right]$
C)
$\left[ \begin{array} { r r r | r } 12 & 10 & 11 & - 36 \\ 1 & 13 & - 3 & 0 \\ 2 & - 7 & 4 & 21 \end{array} \right]$
D)
$$\left[ \begin{array} { r r r | r } 12 & 10 & 11 & - 12 \\\frac { 1 } { 3 } & \frac { 13 } { 3 } & - 1 & 0 \\\frac { 2 } { 3 } & - \frac { 7 } { 3 } & \frac { 4 } { 3 } & 7\end{array} \right]$$

Question 20

Perform the matrix row operation (or operations)and write the new matrix.
$\left[ \begin{array} { r r r | r } 1 & - 4 & 1 & 3 \\ - 5 & 0 & 3 & - 3 \\ - 1 & 2 & - 2 & - 1 \end{array} \right] - 3 R _ { 1 } + R _ { 2 }$

Question 21

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { r } 5 x + 2 y + z = - 11 \\2 x - 3 y - z = 17 \\7 x - y = 12\end{array}$$

A) $\varnothing$
B) $\{ ( 0 , - 6,1 ) \}$
C) $\{ ( - 2,0 , - 1 ) \}$
D) $\{ ( 1 , - 5,0 ) \}$

Question 22

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{aligned}x + y + z & = 9 \\2 x - 3 y + 4 z & = 7 \\x - 4 y + 3 z & = - 2\end{aligned}$$

A) $\left\{ \left( - \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } + \frac { 11 } { 5 } , z \right) \right\}$
B) $\left\{ \left( \frac { z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } + \frac { 11 } { 5 } , z \right) \right\}$
C) $\left\{ \left( - \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } - \frac { 11 } { 5 } , z \right) \right\}$
D) $\left\{ \left( \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } - \frac { 11 } { 5 } , z \right) \right\}$

Question 23

Apply Gaussian Elimination to Systems with More Variables than Equations
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { l } 5 x - y + z = 8 \\7 x + y + z = 6\end{array}$$

A) $\left\{ \left( - \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z - \frac { 13 } { 6 } , z \right) \right\}$
B) $\{ ( - \mathrm { z } + 3,4 \mathrm { z } + 7 , \mathrm { z } ) \}$
C) $\left\{ \left( \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z , z \right) \right\}$
D) $\varnothing$

Question 24

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { r } 4 x - y + 3 z = 12 \\x + 4 y + 6 z = - 32 \\5 x + 3 y + 9 z = 20\end{array}$$

A) $\varnothing$
B) $\{ ( 2 , - 7 , - 1 ) \}$
C) $\{ ( 8 , - 7 , - 2 ) \}$
D) $\{ ( - 8 , - 7,9 ) \}$

Question 25

Apply Gaussian Elimination to Systems with More Variables than Equations
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { r } 3 x + y + z - 2 w = 10 \\2 x + 3 y + 3 z + w = - 5 \\2 x + y + 4 z + 11 w = 11\end{array}$$

A) $\{ ( w + 5,3 w - 7 , - 4 w + 2 , w ) \}$
B) $\{ ( 2 w + 3,6 w - 7 , - 10 w + 8 , w ) \}$
C) $\{ ( 6 , - 4 , - 2,1 ) \}$
D) $\{ ( 7 , - 1 , - 6,2 ) \}$

Question 26

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { l } x - y + z - w = 10 \\\quad - 2 x + 3 y + 5 w = - 28 \\x + 2 y + 8 z + 3 w = - 10 \\x - 4 y - 6 z - 5 w = 30 \\\begin{array} { l l } \text { A) } \{ ( - 17 w - 10 , - 13 w - 16,5 w + 4 , w ) \} & \text { B) } \{ ( 3 w - 2 , - 8 w + 3,4 w + 9 , w ) \} \\\text { C) } \{ ( 24,10 , - 6 , - 2 ) \} & \text { D) } \varnothing\end{array}\end{array}$$

Question 27

Use Matrices and Gauss-Jordan Elimination to Solve Systems
Solve the system of equations using matrices. Use Gauss-Jordan elimination.
$$\begin{aligned}3 x + 5 y + 2 w & = - 12 \\2 x + 6 z - w & = - 5 \\- 2 y + 3 z - 3 w & = - 3 \\- x + 2 y + 4 z + w & = - 2\end{aligned}$$

A) $\{ ( - 1 , - 3,0,3 ) \}$
B) $\{ ( 1 , - 3,0,3 ) \}$
C) $\{ ( - 1,3,0 , - 3 ) \}$
D) $\{ ( 1,3,0 , - 3 ) \}$

Question 28

Write a system of linear equations in three variables, and then use matrices to solve the system.
Ron attends a cocktail party (with his graphing calculator in his pocket). He wants to limit his food intake to 131 g protein, 107 g fat, and 165 g carbohydrate. According to the health conscious hostess, the
Marinated mushroom caps have 3 g protein, 5 g fat, and 9 g carbohydrate; the spicy meatballs have 14 g
Protein, 7 g fat, and 15 g carbohydrate; and the deviled eggs have 13 g protein, 15 g fat, and 6 g
Carbohydrate. How many of each snack can he eat to obtain his goal?

A)7 mushrooms; 6 meatballs; 2 eggs
B)6 mushrooms; 2 meatballs; 7 eggs
C)2 mushrooms; 7 meatballs; 6 eggs
D)8 mushrooms; 7 meatballs; 3 eggs

Question 29

Apply Gaussian Elimination to Systems with More Variables than Equations
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { l } x + y + z = 7 \\x - y + 2 z = 7\end{array}$$

A) $\left\{ \left( - \frac { 3 } { 2 } z + 7 , \frac { 1 } { 2 } z , z \right) \right\}$
B) $\{ ( - 3 z + 14,2 z - 7 , z ) \}$
C) $\{ ( 4,1,2 ) \}$
D) $\{ ( 8 , - 3,2 ) \}$

Question 30

Use Matrices and Gauss-Jordan Elimination to Solve Systems
Solve the system of equations using matrices. Use Gauss-Jordan elimination.
$$\begin{aligned}x + y - z + w & = - 5 \\3 x - y + 3 z - 2 w & = 7 \\- 2 x + 2 y + z - w & = 16 \\- x - 2 y - 3 z + 3 w & = - 22\end{aligned}$$

A) $\{ ( - 2,3,4 , - 2 ) \}$
B) $\left\{ \left( - 2 , - 3,5 , \frac { 1 } { 2 } \right) \right\}$
C) $\{ ( 2 , - 3 , - 4 , - 2 ) \}$
D) $\left\{ \left( \frac { 1 } { 2 } , - \frac { 1 } { 3 } , - \frac { 1 } { 4 } , - \frac { 1 } { 2 } \right) \right\}$

Question 31

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { r r } x + y + z + w = & 7 \\3 x - 2 z + 5 w = & 11 \\- 4 x + 3 y + w = & 4 \\- x - y - z - w = & 6\end{array}$$

A) $\varnothing$
B) $\left\{ \left( \frac { 3 } { 2 } , 1 , \frac { 1 } { 3 } , - 2 \right) \right\}$
C) $\left\{ \left( \frac { 7 } { 4 } , - \frac { 1 } { 2 } , 5 , - \frac { 1 } { 6 } \right) \right\}$
D) $\left\{ \left( - 11 , \frac { 7 } { 19 } , \frac { 6 } { 19 } , - 4 \right) \right\}$

Question 32

Apply Gaussian Elimination to Systems with More Variables than Equations
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { c } x + y + z = 9 \\2 x - 3 y + 4 z = 7\end{array}$$

A) $\left\{ \left( - \frac { 7 } { 5 } z + \frac { 34 } { 5 } , \frac { 2 } { 5 } z + \frac { 11 } { 5 } , z \right) \right\}$
B) $\left\{ \left( \frac { 3 } { 5 } z + \frac { 16 } { 5 } , - \frac { 8 } { 5 } z + \frac { 29 } { 5 } , z \right) \right\}$
C) $\left\{ \left( \frac { 27 } { 5 } , \frac { 13 } { 5 } , 1 \right) \right\}$
D) $\varnothing$

Question 33

Write a system of linear equations in three variables, and then use matrices to solve the system.
The table below shows the number of birds for three selected years after an endangered species protection program was started.

Use the quadratic function $y = a x ^ { 2 } + b x + c$ to model the data. Solve the system of linear equations involving $a , b$ , and $c$ using matrices. Find the equation that models the data.

A) $y = 5 x ^ { 2 } + 12 x + 25$
B) $y = 6 x ^ { 2 } + 24 x + 20$
C) $y = 7 x ^ { 2 } - 12 x + 28$
D) $y = 10 x ^ { 2 } - 36 x + 21$

Question 34

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{aligned}x + 8 y + 8 z & = 8 \\7 x + 7 y + z & = 1 \\8 x + 15 y + 9 z & = - 9\end{aligned}$$

A) $\varnothing$
B) $\{ ( 0,0,1 ) \}$
C) $\{ ( 1 , - 1,1 ) \}$
D) $\{ ( - 1,0,1 ) \}$

Question 35

Write a system of linear equations in three variables, and then use matrices to solve the system.
A ceramics workshop makes wreaths, trees, and sleighs for sale at Christmas. A wreath takes 3 hours to prepare, 2 hours to paint, and 10 hours to fire. A tree takes 15 hours to prepare, 3 hours to paint, and 4
Hours to fire. A sleigh takes 4 hours to prepare, 16 hours to paint, and 7 hours to fire. If the workshop has
93 hours for prep time, 74 hours for painting, and 107 hours for firing, how many of each can be made?

A)7 wreaths; 4 trees; 3 sleighs
B)4 wreaths; 3 trees; 7 sleighs
C)3 wreaths; 7 trees; 4 sleighs
D)8 wreaths; 5 trees; 4 sleighs

Question 36

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { r } 3 x - 2 y + 2 z - w = 2 \\4 x + y + z + 6 w = 8 \\- 3 x + 2 y - 2 z + w = 5 \\5 x + 3 z - 2 w = 1\end{array}$$

A) $\varnothing$
B) $\left\{ \left( 2,0 , - \frac { 3 } { 37 } , \frac { 9 } { 37 } \right) \right\}$
C) $\left\{ \left( \frac { 1 } { 2 } , 0 , - \frac { 37 } { 3 } , \frac { 37 } { 9 } \right) \right\}$
D) $\left\{ \left( 1 , - \frac { 1 } { 3 } , \frac { 4 } { 9 } , 6 \right) \right\}$

Question 37

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { r } x + y + z + w = 8 \\3 x + 2 y + z + 4 w = 21 \\4 x + 4 y + 5 z + 8 w = 30 \\2 x + 3 y + 6 z + 9 w = 15\end{array}$$

A) $\{ ( - 6 w + 3,9 w + 7 , - 4 w - 2 , w ) \}$
B) $\{ ( 5 w + 11 , - 3 w - 7 , - 3 w + 4 , w ) \}$
C) $\{ ( - 3,16 , - 6,1 ) \}$
D) $\varnothing$

Question 38

Write a system of linear equations in three variables, and then use matrices to solve the system.
There were approximately 100,000 vehicles sold at a particular dealership last year. The dealer tracks sales by age group for marketing purposes. The percentage of 36- to 59-year-old buyers and the percentage of
Buyers 60 and older combined exceeds the percentage of buyers 35 and younger by 38%. If the percentage
Of buyers in the oldest group is doubled, it is 24% less than the percentage of users in the middle group.
Find the percentage of buyers in each of the three age groups.

A)31% 35 and younger; 54% 36-59 year olds; 15% 60 and older
B)33% 35 and younger; 51% 36-59 year olds; 16% 60 and older
C)25% 35 and younger; 56% 36-59 year olds; 19% 60 and older
D)15% 35 and younger; 54% 36-59 year olds; 31% 60 and older

Question 39

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { r } x + 3 y + 2 z = 11 \\4 y + 9 z = - 12 \\x + 7 y + 11 z = - 1\end{array}$$

A) $\left\{ \left( \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } - 3 , z \right) \right\}$
B) $\left\{ \left( \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } + 3 , z \right) \right\}$
C) $\left\{ \left( \frac { 19 z } { 4 } + 20 , \frac { 9 z } { 4 } + 3 , z \right) \right\}$
D) $\left\{ \left( - \frac { 19 \mathrm { z } } { 4 } + 20 , - \frac { 9 \mathrm { z } } { 4 } + 3 , \mathrm { z } \right) \right\}$

Question 40

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{array} { r } x + y + z = 7 \\x - y + 2 z = 7 \\2 x + 3 z = 14\end{array}$$

A) $\left\{ \left( - \frac { 3 z } { 2 } + 7 , \frac { z } { 2 } , z \right) \right\}$
B) $\left\{ \left( - \frac { 3 z } { 2 } - 7 , \frac { z } { 2 } , z \right) \right\}$
C) $\left\{ \left( - \frac { 3 z } { 2 } + 7,2 z , z \right) \right\}$
D) $\left\{ \left( - \frac { 3 z } { 2 } - 7,2 z , z \right) \right\}$

Question 41

Solve the problem.
Let $\mathrm { A } = \left[ \begin{array} { r r r } 7 & 2 & 4 \\ 14 & 0 & - 1 \\ 2 & 5 & 3 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } 5 & - 2 & 1 \\ 2 & 0 & 1 \\ - 3 & 6 & - 5 \end{array} \right]$ . Find $\mathrm { A } + \mathrm { B }$ .

A) $\left[ \begin{array} { r r r } 12 & 0 & 5 \\ 16 & 0 & 0 \\ - 1 & 11 & - 2 \end{array} \right]$
B)
$\left[ \begin{array} { r r r } 16 & - 4 & 5 \\ 12 & 0 & 0 \\ - 1 & 11 & - 2 \end{array} \right]$
C)
$\left[ \begin{array} { r r r } 16 & - 4 & 5 \\ 16 & 0 & - 2 \\ 1 & 11 & - 2 \end{array} \right]$
D)
$\left[ \begin{array} { r r r } 12 & 0 & 5 \\ 12 & 0 & - 2 \\ 1 & 11 & - 2 \end{array} \right]$

Question 42

Solve the problem.
Let $A = \left[ \begin{array} { r r r } 2 & 2 & 3 \\ 4 & 0 & - 3 \\ - 4 & 6 & 9 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } 6 & - 2 & 3 \\ - 4 & 0 & 1 \\ 3 & 9 & - 6 \end{array} \right]$ . Find $A = B$ .

A)
$\left[ \begin{array} { r r r } - 4 & 4 & 0 \\ 8 & 0 & - 4 \\ - 7 & - 3 & 15 \end{array} \right]$
B)
$\left[ \begin{array} { r r r } - 4 & 0 & 0 \\ 8 & 0 & - 4 \\ 1 & - 3 & 3 \end{array} \right]$
C)
$\left[ \begin{array} { r r r } - 4 & - 4 & 0 \\ 8 & 0 & - 4 \\ - 7 & 3 & 15 \end{array} \right]$
D)
$\left[ \begin{array} { r r r } 8 & 0 & 6 \\ 0 & 0 & - 2 \\ - 1 & 15 & 3 \end{array} \right]$

Question 43

Give the order of the matrix, and identify the given element of the matrix.
$$\left[ \begin{array} { c c c c } - 1 & 9 & 5 & 13 \\- 13 & 7 & - 15 & - 10\end{array} \right] ; a _ { 12 }$$

A) $2 \times 4 ; 9$
B) $4 \times 2 ; 9$
C) $2 \times 4 ; - 13$
D) $4 \times 2 ; - 13$

Question 44

Solve the problem.
Let $\mathrm { A } = \left[ \begin{array} { l l } 3 & 3 \\ 2 & 5 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 0 & 4 \\ - 1 & 6 \end{array} \right]$ . Find $2 \mathrm {~A} + \mathrm { B }$ .

A)
$\left[ \begin{array} { l l } 6 & 10 \\ 3 & 16 \end{array} \right]$
B)
$\left[ \begin{array} { l l } 6 & 14 \\ 2 & 22 \end{array} \right]$
C)
$\left[ \begin{array} { l l } 6 & 10 \\ 1 & 11 \end{array} \right]$
D)
$\left[ \begin{array} { r r } 6 & 7 \\ 3 & 11 \end{array} \right]$

Question 45

Solve the problem.
Let $A = \left[ \begin{array} { r r } - 6 & 6 \\ - 1 & - 5 \\ 6 & - 8 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 2 & - 5 \\ - 3 & - 9 \\ - 6 & 6 \end{array} \right]$ . Find $A + B$ .

A)
$\left[ \begin{array} { r r } - 4 & 1 \\ - 4 & - 14 \\ 0 & - 2 \end{array} \right]$
B)
$\left[ \begin{array} { r r } - 8 & 11 \\ 2 & 4 \\ 12 & - 17 \end{array} \right]$
C)
$\left[ \begin{array} { r r } - 4 & 1 \\ 4 & - 5 \\ 0 & 2 \end{array} \right]$
D)
$\left[ \begin{array} { r r } - 4 & - 5 \\ - 4 & - 14 \\ 0 & - 2 \end{array} \right]$

Question 46

Solve the problem.
Let $\mathrm { B } = \left[ \begin{array} { l l l l } - 1 & 4 & 5 & - 3 \end{array} \right]$ . Find $- 2 \mathrm {~B}$ .

A) $\left[ \begin{array} { l l l l } 2 & - 8 & - 10 & 6 \end{array} \right]$
$\text { B) }\left[\begin{array}{llll}2 & 4 & 5 & -3\end{array}\right]$

C) $\left[ \begin{array} { l l l l } - 2 & 8 & 10 & - 6 \end{array} \right]$
D) $\left[ \begin{array} { l l l l } - 3 & 2 & 3 & - 5 \end{array} \right]$

Question 47

Understand What is Meant by Equal Matrices
Find values for the variables so that the matrices are equal.
$$\left[ \begin{array} { r r } x + 3 & y + 4 \\7 & 6\end{array} \right] = \left[ \begin{array} { r r } 5 & - 4 \\7 & z\end{array} \right]$$
B) $x = - 2 ; y = 8 ; z = - 6$

A) $x = 2 ; y = - 8 ; z = 6$
D) $x = 2 ; y = 6 ; z = 5$
C) $x = 5 ; y = - 4 ; z = 6$

Question 48

Apply Gaussian Elimination to Systems with More Variables than Equations
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
$$\begin{aligned}2 x + y + 2 z - 4 w & = 10 \\x + 3 y + 2 z - 11 w & = 17 \\3 x + y + 7 z - 21 w & = 0\end{aligned}$$

A) $\{ ( - 3 w + 5,2 w + 6,4 w - 3 , w ) \}$
B) $\{ ( 3 w + 5,6 w + 6 , - 4 w - 3 , w ) \}$
C) $\{ ( w + 5,8 w + 4 , - 3 w - 2 , w ) \}$
D) $\{ ( w - 5,8 w - 4 , - 3 w + 2 , w ) \}$

Question 49

Understand What is Meant by Equal Matrices
Find values for the variables so that the matrices are equal.

A) $x = 5 ; y = - 6 ; z = 3$
B) $x = 5 ; y = - 6 ; z = - 2$
C) $x = 5 ; y = - 2 ; z = 3$
D) $x = - 6 ; y = 5 ; z = 3$

Question 50

Solve Problems Involving Systems Without Unique Solutions
Solve the problem using matrices.
The figure below shows the intersection of three one-way streets. To keep traffic moving, the number of cars per minute entering an intersection must equal the number of cars leaving that intersection. Set up a
System of equations that keeps traffic moving, and use Gaussian elimination to solve the system. If
Construction limits z to t cars per minute, how many cars per minute must pass through the other
Intersections to keep traffic moving?

A) $t + 8$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } + 3 \mathrm { cars } / \mathrm { min }$ between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$
B) $t + 1$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } + 4 \mathrm { cars } / \mathrm { min }$ between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$
C) $t - 2$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } + 1$ cars/min between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$
D) $t + 2$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } - 3 \mathrm { cars } / \mathrm { min }$ between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$

Question 51

Solve Problems Involving Systems Without Unique Solutions
Solve the problem using matrices.
The nutritional content per ounce for three foods is given in the table below.

What combination of these foods can provide exactly 14 grams of fat, 27 grams of protein, and 10 grams of fiber?

A) No possible combination of these foods
B) 3 oz of Food A; 5 oz of Food B; 1 oz of Food $C$
C) $7 \mathrm { oz }$ of Food A; 7 oz of Food B; 1 oz of Food $C$
D) $4 \mathrm { oz }$ of Food $A ; 6 \mathrm { oz }$ of Food B; 2 oz of Food $C$

Question 52

Solve the problem.
Let $\mathrm { A } = \left[ \begin{array} { r r } - 3 & 5 \\ 0 & 2 \end{array} \right]$ . Find $4 \mathrm {~A}$ .

A)
$\left[ \begin{array} { r r } - 12 & 20 \\ 0 & 8 \end{array} \right]$
B)
$\left[ \begin{array} { r r } - 12 & 20 \\ 0 & 2 \end{array} \right]$
C)
$\left[ \begin{array} { r } - 125 \\ 02 \end{array} \right]$
D)
$\left[ \begin{array} { l l } 1 & 9 \\ 4 & 6 \end{array} \right]$

Question 53

Give the order of the matrix, and identify the given element of the matrix.
$$\left[ \begin{array} { c c c c c } 0 & - 15 & - 7 & 13 & 7 \\5 & - e & - 5 & 14 & \pi \\- 6 & 7 & 2 & 1 & - 7 \\\frac { 1 } { 3 } & - 6 & - 8 & 2 & 15\end{array} \right] ; a _ { 34 }$$

A) $4 \times 5 ; 1$
B) $5 \times 4 ; - 8$
C) $20 ; - 7$
D) $4 \times 4 ; 2$

Question 54

Understand What is Meant by Equal Matrices
Find values for the variables so that the matrices are equal.
$$\left[ \begin{array} { r r } x & y + 5 \\7 z & 10\end{array} \right] = \left[ \begin{array} { c c } 3 & 11 \\56 & 10\end{array} \right]$$

A) $x = 3 ; y = 6 ; z = 8$
B) $x = 3 ; y = 11 ; z = 56$
C) $x = 11 ; y = 10 ; z = 3$
D) $x = 10 ; y = 16 ; z = 392$

Question 55

Solve the problem.
Let $A = \left[ \begin{array} { r } 3 \\ - 5 \\ - 4 \end{array} \right]$ and $B = \left[ \begin{array} { r } - 5 \\ 6 \\ 5 \end{array} \right]$ . Find $A + B$ .

A)
$\left[\begin{array}{r}-2 \\1 \\1\end{array}\right]$

B)
$\left[ \begin{array} { l l l } - 2 & 1 & 1 \end{array} \right]$
C)
$\left[\begin{array}{rr}3 & -5 \\-5 & 6 \\-4 & 5\end{array}\right]$


D)
$\left[\begin{array}{l}2 \\1 \\2\end{array}\right]$

Question 56

Solve the problem.
Let $A = \left[ \begin{array} { r r } - 7 & 1 \\ 2 & 5 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 6 & 2 \\ 3 & - 3 \end{array} \right]$ . Find $A + B$ .

A)
$\left[ \begin{array} { r r } - 1 & 3 \\ 5 & 2 \end{array} \right]$
B)
$\left[ \begin{array} { r l } 3 & 4 \\ - 2 & 2 \end{array} \right]$
C)
$\left[ \begin{array} { r r } - 1 & - 5 \\ - 4 & - 10 \end{array} \right]$

D)
$[ 9 ]$

Question 57

Understand What is Meant by Equal Matrices
Find values for the variables so that the matrices are equal.
$$\left[ \begin{array} { l } x \\9\end{array} \right] = \left[ \begin{array} { l } 1 \\y\end{array} \right]$$

A) $x = 1 ; y = 9$
B) $x = 9 ; y = 1$
C) $x = 1 ; y = 1$
D) $x = 9 ; y = 9$

Question 58

Solve the problem.
Let $A = \left[ \begin{array} { r r } - 1 & 4 \\ 0 & 4 \\ 9 & - 4 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 7 & 2 \\ 17 & 4 \\ 3 & 2 \end{array} \right]$ . Find $A - B$ .

A)
$\left[ \begin{array} { r r } - 8 & 2 \\ - 17 & 0 \\ 6 & - 6 \end{array} \right]$
B)
$\left[ \begin{array} { r r } 1 & 5 \\ 7 & 8 \\ 12 & 0 \end{array} \right]$
C)
$\left[ \begin{array} { r r } 1 & 2 \\ 7 & 0 \\ 6 & - 2 \end{array} \right]$
D)
$\left[ \begin{array} { r r } 3 & - 3 \\ 7 & 0 \\ - 6 & 6 \end{array} \right]$

Question 59

Solve the problem.
Let $A = \left[ \begin{array} { r r } - 1 & 0 \\ 5 & 3 \end{array} \right]$ and $B = \left[ \begin{array} { r r } - 1 & 5 \\ 3 & 1 \end{array} \right]$ . Find $A - B$ .

A)
$\left[ \begin{array} { r r } 0 & - 5 \\ 2 & 2 \end{array} \right]$
B)
$\left[\begin{array}{rr}-2 & 5 \\8 & 4\end{array}\right]$
C)
$\left[ \begin{array} { r r } 0 & 5 \\ - 2 & - 2 \end{array} \right]$
D)
$[ - 1 ]$

Question 60

Solve Problems Involving Systems Without Unique Solutions
Solve the problem using matrices.
A company that manufactures products A, B, and C does both assembly and testing. The hours needed to assemble and test each product are shown in the table below.

The company has exactly 24 hours per week available for assembly and 109 hours per week available for testing. If the company must produce $t$ units of Product $C$ this week, how many units of Products $A$ and $B$ can they produce?

A) 11 of Product A; $- 2 t + 13$ of Product B
B) 11t of Product A; 2t $+ 13$ of Product B
C) $t + 11$ of Product $A ; t + 13$ of Product $B$
D) 11 of Product $A ; 13$ of Product $B$

Question 61

Solve the problem.
Let $A = \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]$ and $B = \left[ \begin{array} { r } - 1 \\ 3 \\ - 2 \end{array} \right]$ . Find $A - 2 B$

A)
$\left[ \begin{array} { r } 3 \\ - 9 \\ 6 \end{array} \right]$
B)
$\left[ \begin{array} { r } - 1 \\ 3 \\ - 2 \end{array} \right]$
C)
$\left[ \begin{array} { r } - 3 \\ 9 \\ - 6 \end{array} \right]$
D)
$\left[ \begin{array} { r } 3 \\ - 6 \\ 4 \end{array} \right]$

Question 62

Find the product AB, if possible.
$A = \left[ \begin{array} { l l l } - 5 & 2 & 1 \end{array} \right] , B = \left[ \begin{array} { r r r } - 6 & - 8 & 3 \\ - 3 & 7 & - 4 \\ - 9 & 2 & 5 \end{array} \right]$

A) $[ 1556 - 18 ]$
B)
$\left[ \begin{array} { r } 15 \\ 56 \\ - 18 \end{array} \right]$
C)
$$\left[ \begin{array} { r r r } - 5 & 2 & 1 \\- 6 & - 8 & 3 \\- 3 & 7 & - 4 \\- 9 & 2 & 5\end{array} \right] \quadD) \left[ \begin{array} { r r r } 30 & - 16 & 3 \\15 & 14 & - 4 \\45 & 4 & 5\end{array} \right]$$

Question 63

Solve the matrix equation for X.
Let $A = \left[ \begin{array} { r r } 1 & 4 \\ 0 & - 2 \\ 6 & - 6 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 8 & - 6 \\ - 1 & 4 \\ 0 & 5 \end{array} \right] ; \quad B - X = 3 A$

A)
$X = \left[ \begin{array} { r r } 5 & - 18 \\ - 1 & 10 \\ - 18 & 23 \end{array} \right]$
B)
$X = \left[ \begin{array} { r r } 11 & 6 \\ 1 & - 2 \\ 18 & - 13 \end{array} \right]$
C)
$X = \left[ \begin{array} { r r } 5 & - 18 \\ 1 & 10 \\ - 18 & 23 \end{array} \right]$
D)
$X = \left[ \begin{array} { r r } 11 & 6 \\ 1 & - 2 \\ 6 & - 13 \end{array} \right]$

Question 64

Model Applied Situations with Matrix Operations
The $\perp$ shape in the figure below is shown using 9 pixels in a $3 \times 3$ grid. The color levels are given to the right of the figure. Use the matrix $\left[ \begin{array} { l l l } 1 & 3 & 1 \\ 1 & 3 & 1 \\ 3 & 3 & 3 \end{array} \right]$ that represents a digital photograph of the $\perp$ shape to solve the problem.
Adjust the contrast by changing the black to dark grey and the light grey to white. Use matrix addition to accomplish this. A)
$$\left[ \begin{array} { l l l } 1 & 3 & 1 \\1 & 3 & 1 \\3 & 3 & 3\end{array} \right] + \left[ \begin{array} { l l l } - 1 & - 1 & - 1 \\- 1 & - 1 & - 1 \\- 1 & - 1 & - 1\end{array} \right] = \left[ \begin{array} { l l l } 0 & 2 & 0 \\0 & 2 & 0 \\2 & 2 & 2\end{array} \right]$$
B)
$$\left[ \begin{array} { l l l } 1 & 3 & 1 \\1 & 3 & 1 \\3 & 3 & 3\end{array} \right] + \left[ \begin{array} { r r r } 0 & - 1 & 0 \\0 & - 1 & 0 \\- 1 & - 1 & - 1\end{array} \right] = \left[ \begin{array} { l l l } 1 & 2 & 1 \\1 & 2 & 1 \\2 & 2 & 2\end{array} \right]$$
C)
$$\left[ \begin{array} { l l l } 1 & 3 & 1 \\1 & 3 & 1 \\3 & 3 & 3\end{array} \right] + \left[ \begin{array} { l l l } 1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1\end{array} \right] = \left[ \begin{array} { l l l } 2 & 4 & 2 \\2 & 4 & 2 \\4 & 4 & 4\end{array} \right]$$
D)
$$\left[ \begin{array} { l l l } 1 & 3 & 1 \\1 & 3 & 1 \\3 & 3 & 3\end{array} \right] + \left[ \begin{array} { r r r } 0 & - 1 & 0 \\0 & - 1 & 0 \\- 1 & - 1 & - 1\end{array} \right] = \left[ \begin{array} { l l l } 0 & 2 & 0 \\0 & 2 & 0 \\2 & 2 & 2\end{array} \right]$$

Question 65

Find the product AB, if possible.
$A = \left[ \begin{array} { l l l } - 9 & - 4 & - 4 \\ - 4 & - 9 & - 1 \end{array} \right] , B = \left[ \begin{array} { l } - 1 \\ - 3 \\ - 9 \end{array} \right]$

A)
$\left[ \begin{array} { l } 57 \\ 40 \end{array} \right]$
B) $\mathrm { AB }$ is not defined.
C) $\left[ \begin{array} { l l } 57 & 40 \end{array} \right]$
D)

$\left[ \begin{array} { c c c } - 9 & - 4 & - 4 \\ - 4 & - 9 & - 1 \\ - 1 & - 3 & - 9 \end{array} \right]$

Question 66

Find the product AB, if possible.
$A = \left[ \begin{array} { l l l } - 3 & 2 & 9 \end{array} \right] , \quad B = \left[ \begin{array} { r } 7 \\ 0 \\ - 3 \end{array} \right]$

A) $[ - 48 ]$
B) $[ 183 ]$
C) $\left[ \begin{array} { l l l } - 21 & 0 & - 27 \end{array} \right]$
D)
$\left[\begin{array}{r}-21 \\0 \\-27\end{array}\right]$

Question 67

Solve the matrix equation for X.
Let $A = \left[ \begin{array} { r r } 5 & - 3 \\ 1 & 1 \\ 4 & 1 \end{array} \right]$ and $B = \left[ \begin{array} { l l } - 3 & - 7 \\ - 3 & - 6 \\ - 6 & - 3 \end{array} \right] ; \quad X - B = A$

A)
$\left[ \begin{array} { r r } 2 & - 10 \\ - 2 & - 5 \\ - 2 & - 2 \end{array} \right]$
B)
$\left[ \begin{array} { r } 84 \\ 47 \\ 102 \end{array} \right]$
C)
$\left[ \begin{array} { r r } 2 & - 10 \\ 2 & 1 \\ - 2 & 2 \end{array} \right]$
D)
$\left[ \begin{array} { r r } 2 & 1 \\ - 2 & - 5 \\ - 2 & - 2 \end{array} \right]$

Question 68

Solve the matrix equation for X.
Let $\mathrm { A } = \left[ \begin{array} { r r } 3 & 3 \\ - 3 & 1 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 3 & 2 \\ 3 & - 2 \end{array} \right] ; \quad \mathrm { X } + \mathrm { A } = \mathrm { B }$

A)
$X = \left[ \begin{array} { r r } - 6 & - 1 \\ 6 & - 3 \end{array} \right]$
B)
$X = \left[ \begin{array} { r r } - 1 & - 6 \\ - 3 & 6 \end{array} \right]$
C)
$$X = \left[ \begin{array} { r r } - 3 & 6 \\- 1 & - 6\end{array} \right]$$

D)
$X=\left[\begin{array}{rr}-3 & 6 \\-1 & -6\end{array}\right]$

Question 69

Find the product AB, if possible.
$\mathrm { A } = \left[ \begin{array} { r r } - 2 & 3 \\ 3 & 2 \end{array} \right] , \mathrm { B } = \left[ \begin{array} { l l } - 2 & 0 \\ - 1 & 3 \end{array} \right]$

A)
$\left[ \begin{array} { r r } 1 & 9 \\ - 8 & 6 \end{array} \right]$
B)
$\left[ \begin{array} { r r } 4 & 0 \\ - 3 & 6 \end{array} \right]$
C)
$\left[\begin{array}{rr}4 & -6 \\-4 & 3\end{array}\right]$
D)
$\left[ \begin{array} { r r } 9 & 1 \\ 6 & - 8 \end{array} \right]$

Question 70

Find the product AB, if possible.
$\mathrm { A } = \left[ \begin{array} { r r } - 1 & 3 \\ 1 & 6 \end{array} \right] , \mathrm { B } = \left[ \begin{array} { l l l } 0 & - 2 & 4 \\ 1 & - 3 & 2 \end{array} \right]$

A) $\left[ \begin{array} { r r r } 3 & - 7 & 2 \\ 6 & - 20 & 16 \end{array} \right]$
B) $\mathrm { AB }$ is not defined.
C)
$\left[ \begin{array} { r r } 3 & 6 \\ - 7 & - 20 \\ 2 & 16 \end{array} \right]$
D)
$\left[ \begin{array} { r r r } 0 & - 6 & 12 \\ 1 & - 18 & 12 \end{array} \right]$

Question 71

Find the product AB, if possible.
$$A = \left[ \begin{array} { r r r } 4 & 4 & - 6 \\- 3 & - 9 & 5 \\9 & 9 & - 9\end{array} \right] , B = \left[ \begin{array} { r r r } - 4 & - 3 & 4 \\6 & - 4 & - 7 \\5 & - 2 & 1\end{array} \right]$$

A)
$\left[ \begin{array} { r r r } - 22 & - 16 & - 18 \\ - 17 & 35 & 56 \\ - 27 & - 45 & - 36 \end{array} \right]$
B)
$\left[ \begin{array} { r r r } - 22 & - 17 & - 27 \\ - 16 & 35 & - 45 \\ - 18 & 56 & - 36 \end{array} \right]$
C)

$\left[ \begin{array} { r r r } 4 & 4 & - 6 \\ - 3 & - 9 & 5 \\ 9 & 9 & - 9 \\ - 4 & - 3 & 4 \\ 6 & - 4 & - 7 \\ 5 & - 2 & 1 \end{array} \right]$
D)

$\left[ \begin{array} { r r r } - 16 & - 12 & - 24 \\ - 18 & 36 & - 35 \\ 45 & - 18 & - 9 \end{array} \right]$

Question 72

Solve the matrix equation for X.
Let $A = \left[ \begin{array} { r r } 3 & - 3 \\ - 4 & 0 \\ 4 & - 5 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 4 & 0 \\ 0 & - 2 \\ 3 & - 5 \end{array} \right]$

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

D)
$X=\left[\begin{array}{rr}-1 & 3 \\-4 & 2 \\1 & 0\end{array}\right]$

Question 73

Find the product AB, if possible.
$$A = \left[ \begin{array} { r r r } 1 & 3 & - 2 \\2 & 0 & 5\end{array} \right] , B = \left[ \begin{array} { r r } 3 & 0 \\- 2 & 1 \\0 & 5\end{array} \right]$$

A)
$\left[ \begin{array} { r r } - 7 & - 3 \\ 25 & 6 \end{array} \right]$
B) $\mathrm { AB }$ is not defined.
C)
$\left[ \begin{array} { r r } - 3 & - 7 \\ 6 & 25 \end{array} \right]$
D)
$\left[ \begin{array} { r r r } 3 & - 6 & 0 \\ 0 & 0 & 25 \end{array} \right]$

Question 74

Solve the problem.
Let $A = \left[ \begin{array} { r r r } - 5 & 4 & 4 \\ 4 & 7 & 8 \\ 5 & - 6 & 3 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } 8 & 7 & - 2 \\ - 5 & - 9 & - 8 \\ - 6 & 7 & - 2 \end{array} \right]$ . Find $- 4 A - 3 B$ .

A)
$\left[ \begin{array} { r r r } - 4 & - 37 & - 10 \\ - 1 & - 1 & - 8 \\ - 2 & 3 & - 6 \end{array} \right]$
B)
$\left[ \begin{array} { r r r } 28 & - 9 & - 18 \\ - 21 & - 37 & - 40 \\ - 26 & 31 & - 14 \end{array} \right]$
C)
$\left[ \begin{array} { r r r } 3 & 11 & 2 \\ - 1 & - 2 & 0 \\ - 1 & 1 & 1 \end{array} \right]$
D)
$\left[ \begin{array} { r r r } 3 & - 1 & - 1 \\ 11 & - 2 & 1 \\ 2 & 0 & 1 \end{array} \right]$

Question 75

Find the product AB, if possible.
$A = \left[ \begin{array} { r r r } 3 & - 2 & 1 \\ 0 & 4 & - 3 \end{array} \right] , B = \left[ \begin{array} { r r } 3 & 0 \\ - 2 & 2 \end{array} \right]$

A) $\mathrm { AB }$ is not defined.
B)
$\left[ \begin{array} { r r r } 9 & - 6 & 3 \\ - 6 & 12 & - 8 \end{array} \right]$
C)
$\left[ \begin{array} { r r } 9 & - 6 \\ - 6 & 12 \\ 3 & - 8 \end{array} \right]$
D)
$\left[ \begin{array} { l l } 9 & 0 \\ 0 & 8 \end{array} \right]$

Question 76

Solve the matrix equation for X.
Let $\mathrm { A } = \left[ \begin{array} { r r } 1 & 3 \\ - 1 & - 3 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 1 & - 3 \\ 1 & - 4 \end{array} \right] ; \quad \mathrm { X } + \mathrm { A } = \mathrm { B }$

A)
$X=\left[\begin{array}{rr}-2 & -6 \\2 & -1\end{array}\right]$

B)
$X = \left[ \begin{array} { r r } - 6 & - 2 \\ - 1 & 2 \end{array} \right]$
C)
$X = \left[ \begin{array} { r r } 2 & - 1 \\ - 2 & - 6 \end{array} \right]$
D)
$$X = \left[ \begin{array} { r r } - 1 & - 2 \\- 6 & 2\end{array} \right]$$

Question 77

Find the product AB, if possible.
$A = \left[ \begin{array} { r r r } 3 & - 2 & 1 \\ 0 & 4 & - 3 \end{array} \right] , B = \left[ \begin{array} { r r } 4 & 0 \\ - 2 & 3 \end{array} \right]$

A) $\mathrm { AB }$ is not defined.
B)
$\left[ \begin{array} { r r r } 12 & - 8 & 4 \\ - 6 & 16 & - 11 \end{array} \right]$
C)
$\left[ \begin{array} { r r } 12 & - 6 \\ - 8 & 16 \\ 4 & - 11 \end{array} \right]$
D)

$\left[ \begin{array} { r r } 12 & 0 \\ 0 & 12 \end{array} \right]$

Question 78

Solve the problem.
Let $A = \left[ \begin{array} { l l } - 1 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]$ . Find $2 A + 3 B$ .

A) $[ 14 ]$
B) $[ - 24 ]$
C) $\left[ \begin{array} { l l } - 1 & 4 \end{array} \right]$
D) $\left[ \begin{array} { l l } 2 & 2 \end{array} \right]$

Question 79

Find the product AB, if possible.

A)
B) $\mathrm { AB }$ is not defined.
C) $\left[ \begin{array} { l l } - 21 & 22 \end{array} \right]$
D)
$\left[ \begin{array} { r } - 21 \\ 22 \end{array} \right]$
$\left[ \begin{array} { r r r } - 2 & 7 & - 7 \\ 5 & 9 & - 5 \\ 7 & - 2 & - 1 \end{array} \right]$

Question 80

Solve the matrix equation for X.
Let $\mathrm { A } = \left[ \begin{array} { r r r } 5 & 1 & - 4 \\ 5 & 0 & 0 \\ 1 & - 5 & 4 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } - 1 & - 5 & - 4 \\ 0 & 1 & 1 \\ 5 & 0 & 5 \end{array} \right] ; \quad 4 \mathrm {~B} - 4 \mathrm {~A} = \mathrm { X }$

A)
$$X = \left[ \begin{array} { r r r } - 24 & - 24 & 0 \\- 20 & 4 & 4 \\16 & 20 & 4\end{array} \right]$$
B)
$X = \left[ \begin{array} { r r r } - 20 & 4 & 4 \\ 16 & 20 & 4 \\ - 24 & - 24 & 0 \end{array} \right]$
C)
$X = \left[ \begin{array} { r r r } - 24 & - 24 & 0 \\ - 20 & 1 & 1 \\ 16 & 20 & 4 \end{array} \right]$
D)
$X = \left[ \begin{array} { r r r } - 20 & 1 & 1 \\ 16 & 20 & 4 \\ - 24 & - 24 & 0 \end{array} \right]$