Deck 9: Matrices and Determinants

Question 1

Use Matrices and Gaussian Elimination to Solve Systems
$$\begin{aligned}x + y + z & = 2 \\x - y + 5 z & = - 4 \\2 x + y + z & = 7\end{aligned}$$

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

Answer 1

A

Question 2

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.
$\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & - 3 \\ 0 & 1 & - 4 & 4 & 0 \\ 0 & 0 & 1 & 6 & 17 \\ 0 & 0 & 0 & 1 & 7 \end{array} \right]$

A) $\{ ( 93 , - 128 , - 25,7 ) \}$
В) $\{ ( - 3,0,17,7 ) \}$
C) $\{ ( - 5 , - 1,10,6 ) \}$
D) $\{ ( 7 , - 25 , - 128,93 ) \}$

Answer 2

A

Question 3

Perform Matrix Row Operations
$\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right] - 5 R _ { 1 } + R _ { 2 }$

A)
$\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \\ - 30 & 20 & - 3 & - 23 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]$
B)
$\left[ \begin{array} { c r r | c } 5 & - 4 & 1 & 4 \\ 20 & - 20 & 7 & 17 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]$
C)
$\left[ \begin{array} { r r r | r } 30 & - 4 & - 9 & 19 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]$
D)
$\left[ \begin{array} { c r r | r } - 30 & 20 & - 3 & - 23 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]$

Answer 3

A

Question 4

Simplify Complex Rational Expressions
$$\begin{array} { r } 9 x + 5 z = 65 \\3 y + 6 z = 27 \\2 x + 8 y + 6 z = 42\end{array}$$

A)
$\left[ \begin{array} { l l l | l } 9 & 0 & 5 & 65 \\ 0 & 3 & 6 & 27 \\ 2 & 8 & 6 & 42 \end{array} \right]$

B)

$\left[ \begin{array} { l l | l | l } 9 & 0 & 2 & 65 \\ 0 & 3 & 8 & 27 \\ 5 & 6 & 6 & 42 \end{array} \right]$
C)
$\left[\begin{array}{lll|l}9 & 5 & 0 & 65 \\3 & 6 & 0 & 27 \\2 & 8 & 6 & 42\end{array}\right]$
D)
$\left[\begin{array}{lll}9 & 0 & 5 \\0 & 3 & 6 \\2 & 8 & 6\end{array}\right]$

Question 5

Simplify Complex Rational Expressions
$$\begin{array} { r } x - 4 y + z = 15 \\y + 8 z = 11 \\z = 14\end{array}$$

A)
$\left[ \begin{array} { c c c | c } 1 & - 4 & 1 & 15 \\ 0 & 1 & 8 & 11 \\ 0 & 0 & 1 & 14 \end{array} \right]$
B)

$\left[ \begin{array} { c c | c | c } 0 & - 4 & 0 & 15 \\ 0 & 0 & 8 & 11 \\ 0 & 0 & 0 & 14 \end{array} \right]$
C)
$$\left[ \begin{array} { l l l | l } 1 & 4 & 1 & 15 \\0 & 1 & 8 & 11 \\0 & 0 & 1 & 14\end{array} \right] \quadD) \left[ \begin{array} { r r r | r } 1 & - 4 & 1 & 15 \\1 & 1 & 8 & 11 \\1 & 1 & 14\end{array} \right]$$

Question 6

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables.
$\left[ \begin{array} { r r r | r } 6 & 5 & 3 & - 2 \\ 2 & 0 & 7 & 4 \\ 4 & 2 & 0 & 2 \end{array} \right]$

A) $6 x + 5 y + 3 z = - 2$
$2 x + 7 z = 4$
$4 x + 2 y = 2$
B) $6 x - 5 y + 3 z = - 2$
$2 x + 7 z = - 4$
$4 x + 2 y = - 2$
C) $6 x + 5 y + 3 z = - 2$
$2 x + 7 z = 4$
$4 x + 2 z = 2$
D) $6 x + 5 y + 3 z = - 2$
$2 x + y + 7 z = 4$
$4 x + 2 y + z = 2$
$4 x + 2 y + z = 2$

Question 7

Use Matrices and Gauss-Jordan Elimination to Solve Systems
$$\begin{array} { l } x = - 1 - y - z \\x - y + 4 z = - 6 \\3 x + y = 7 - z\end{array}$$

A) $\{ ( 4 , - 2 , - 3 ) \}$
В) $\{ ( - 2 , - 3,4 ) \}$
C) $\{ ( - 3 , - 2,4 ) \}$
D) $\{ ( - 3,4 , - 2 ) \}$

Question 8

Use Matrices and Gaussian Elimination to Solve Systems
$$\begin{array} { l r } x + y + z - w = & 6 \\2 x - y + 3 z + 4 w = & - 4 \\4 x + 2 y - z - w = & - 13 \\- x - 2 y + 4 z + 3 w = & 12\end{array}$$

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

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.
$\left[ \begin{array} { r r r | r } 1 & 5 & - 8 & 4 \\ 0 & 1 & - 6 & - 4 \\ 0 & 0 & 1 & - 7 \end{array} \right]$

A) $\{ ( 178 , - 46 , - 7 ) \}$
B) $\{ ( 4 , - 4 , - 7 ) \}$
C) $\{ ( 6,1 , - 8 ) \}$
D) $\{ ( 250,38 , - 7 ) \}$

Question 10

Use Matrices and Gauss-Jordan Elimination to Solve Systems
$$\begin{array} { r } - 4 x + 7 y - z = 39 \\x + 6 y + 9 z = 62 \\- 5 x + y + z = - 1\end{array}$$

A) $\{ ( 2,7,2 ) \}$
B) $\{ ( 2,2,7 ) \}$
C) $\{ ( - 2,7,4 ) \}$
D) $\{ ( 4,7 , - 2 ) \}$

Question 11

Perform Matrix Row Operations
$$\left[ \begin{array} { r r r | r } - 20 & 60 & 10 & 45 \\1 & 13 & - 3 & 0 \\2 & - 7 & 4 & 21\end{array} \right] \frac { 1 } { 5 } \mathrm { R } _ { 1 }$$

A)
$$\left[ \begin{array} { c c r | r } - 4 & 12 & 2 & 9 \\1 & 13 & - 3 & 0 \\2 & - 7 & 4 & 21\end{array} \right]$$
B)

$$\left[ \begin{array} { c c r | r } - 20 & 60 & 10 & 45 \\\frac { 1 } { 5 } & \frac { 13 } { 5 } & - \frac { 3 } { 5 } & 0 \\2 & - 7 & 4 & 21\end{array} \right]$$
C)
$$\left[ \begin{array} { r r r | r } - 4 & 12 & 2 & 45 \\1 & 13 & - 3 & 0 \\2 & - 7 & 4 & 21\end{array} \right] \quadD) \left[ \begin{array} { r r r | r } - 4 & 12 & 2 & 9 \\\frac { 1 } { 5 } & \frac { 13 } { 5 } & - \frac { 3 } { 5 } & 0 \\\frac { 2 } { 5 } - \frac { 7 } { 5 } & \frac { 4 } { 5 } & \frac { 21 } { 5 }\end{array} \right]$$

Question 12

Use Matrices and Gauss-Jordan Elimination to Solve Systems
$$\begin{aligned}- 3 x - y - 7 z & = - 76 \\3 x + 3 y - 6 z & = - 27 \\- 9 x - 2 y + z & = - 23\end{aligned}$$

A) $\{ ( 2,7,9 ) \}$
B) $\{ ( 2,9,7 ) \}$
C) $\{ ( - 2,7,4 ) \}$
D) $\{ ( 4,7 , - 2 ) \}$

Question 13

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.
$\left[ \begin{array} { r r r | r } 1 & \frac { 3 } { 2 } & 1 & \frac { 1 } { 2 } \\ 0 & 1 & - \frac { 1 } { 2 } & - 8 \\ 0 & 0 & 1 & 4 \end{array} \right]$

A) $\left\{ \left( \frac { 11 } { 2 } , - 6,4 \right) \right\}$
B) $\left\{ \left( \frac { 1 } { 2 } , - 8,4 \right) \right\}$
C) $\left\{ \left( - 3 , - \frac { 17 } { 2 } , 3 \right) \right\}$
D) $\left\{ \left( - \frac { 21 } { 2 } , - 6,4 \right) \right\}$

Question 14

Simplify Complex Rational Expressions
$$\begin{array} { r } 12 x + 7 y - 8 z + w = 1 \\5 y + z = - 6 \\x - y - 2 z = - 5 \\3 x - 3 y + 4 z = 5\end{array}$$

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

Question 15

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables.
$$\left[ \begin{array} { r r r r | r } 5 & 1 & 0 & 2 & 12 \\- 1 & 8 & 1 & 0 & - 6 \\9 & 0 & 0 & 6 & - 11 \\0 & 3 & 0 & - 3 & - 4\end{array} \right]$$

A)
$5 x + y + 2 w = 12$
$- x + 8 y + z = - 6$
$9 x + 6 w = - 11$
$3 y - 3 w = - 4$
B)
$5 x + y + z + 2 w = 12$
$- x + 8 y + z + w = - 6$
$9 x + y + z + 6 w = - 11$
$x + 3 y + z - 3 w = - 4$
C)
$5 x + y + 2 w = 12$
$x + 8 y + z = - 6$
$9 x + 6 w = - 11$
$3 y + 3 w = - 4$
D)
$5 x + y + 2 z = 12$
$- x + 8 y + z = - 6$
$9 x + 6 y = - 11$
$3 x - 3 y = - 4$

Question 16

Use Matrices and Gaussian Elimination to Solve Systems
$$\begin{aligned}x - y + 4 z & = 15 \\2 x + z & = 3 \\x + 2 y + z & = - 3\end{aligned}$$

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

Question 17

Perform Matrix Row Operations
$\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 3 \\ 0 & - 4 & 3 & - 5 & 0 \\ 3 & 0 & - 1 & - 4 & 1 \\ - 3 & 3 & 0 & 1 & - 4 \end{array} \right] \quad \begin{array} { r } - 2 R _ { 1 } + R _ { 3 } \\ 3 R _ { 1 } + R _ { 4 } \end{array}$

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

Question 18

Use Matrices and Gaussian Elimination to Solve Systems
$$\begin{aligned}5 x - y - 3 z & = - 8 \\3 x - 8 z & = - 21 \\7 y + z & = 31\end{aligned}$$

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

Question 19

Simplify Complex Rational Expressions
$$\begin{array} { l } 2 x + 6 y + 9 z = 66 \\7 x + 6 y + 8 z = 77 \\9 x + 8 y - 2 z = 51\end{array}$$

A)
$\left[ \begin{array} { r r r | r } 2 & 6 & 9 & 66 \\ 7 & 6 & 8 & 77 \\ 9 & 8 & - 2 & 51 \end{array} \right]$

B)

$\left[ \begin{array} { r r r | r } 2 & 7 & 9 & 66 \\ 6 & 6 & 8 & 77 \\ 9 & 8 & - 2 & 51 \end{array} \right]$
C)
$\left[\begin{array}{rrr|r}66 & 9 & 6 & 2 \\77 & 8 & 6 & 7 \\51 & -2 & 8 & 9\end{array}\right]$

D)
$\left[\begin{array}{rrr}2 & 6 & 9 \\7 & 6 & 8 \\9 & 8 & -2\end{array}\right]$

Question 20

Use Matrices and Gaussian Elimination to Solve Systems
$$\begin{aligned}3 x + 5 y - 2 w = & - 13 \\2 x + 7 z - w = & - 1 \\4 y + 3 z + 3 w = & 1 \\- x + 2 y + 4 z = & - 5\end{aligned}$$

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 21

Use Matrices and Gauss-Jordan Elimination to Solve Systems
$$\begin{array} { c } x + y - z + w = - 5 \\3 x - y + 3 z - 2 w = 7 \\- 2 x + 2 y + z - w = 16 \\- x - 2 y - 3 z + 3 w = - 22\end{array}$$

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 22

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
$$\begin{aligned}x + y + z + w & = 7 \\3 x - 2 z + 5 w & = 11 \\- 4 x + 3 y + w & = 4 \\- x - y - z - w & = 6\end{aligned}$$

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 23

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
$$\begin{aligned}4 x - y + 3 z & = 12 \\x + 4 y + 6 z & = - 32 \\5 x + 3 y + 9 z & = 20\end{aligned}$$

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

Question 24

Apply Gaussian Elimination to Systems with More Variables than Equations
$$\begin{array} { l } x + y + z = 9 \\2 x - 3 y + 4 z = 7\end{array}$$

A) $\left\{ \left( - \frac { 7 } { 5 } \mathrm { z } + \frac { 34 } { 5 } , \frac { 2 } { 5 } \mathrm { z } + \frac { 11 } { 5 } , \mathrm { z } \right) \right\}$
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 25

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 103 g protein, 93 g fat, and 135 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; 4 meatballs; 2 eggs
B) 4 mushrooms; 2 meatballs; 7 eggs
C) 2 mushrooms; 7 meatballs; 4 eggs
D) 8 mushrooms; 5 meatballs; 3 eggs

Question 26

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
$$\begin{aligned}5 x + 2 y + z & = - 11 \\2 x - 3 y - z & = 17 \\7 x - y & = 12\end{aligned}$$

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

Question 27

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
$$\begin{array} { r } 3 x - 2 y + 2 z - w = 2 \\4 x + y + z + 6 w = 8 \\- 3 x + 2 y - 2 z + w = 5 \\5 x + 3 z - 2 w = 1\end{array}$$

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 28

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
$$\begin{array} { r } x + 3 y + 2 z = 11 \\4 y + 9 z = - 12 \\x + 7 y + 11 z = - 1\end{array}$$

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) $\left\{ \left( - \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } + 3 , z \right) \right\}$

Question 29

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
$$\begin{aligned}x + 8 y + 8 z & = 8 \\7 x + 7 y + z & = 1 \\8 x + 15 y + 9 z & = - 9\end{aligned}$$

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

Question 30

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 14 hours to prepare, 3 hours to paint, and 4 hours to fire. A sleigh takes 4 hours to prepare, 15 hours to paint, and 7 hours to fire. If the workshop has 85 hours for prep time, 56 hours for painting, and 100 hours for firing, how many of each can be made?

A) 7 wreaths; 4 trees; 2 sleighs
B) 4 wreaths; 2 trees; 7 sleighs
C) 2 wreaths; 7 trees; 4 sleighs
D) 8 wreaths; 5 trees; 3 sleighs

Question 31

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 40%. If the percentage of buyers in the oldest group is doubled, it is 34% less than the percentage of users in the middle group. Find the percentage of buyers in each of the three age groups.

A) 30% 35 and younger; 58% 36-59 year olds; 12% 60 and older
B) 32% 35 and younger; 55% 36-59 year olds; 13% 60 and older
C) 24% 35 and younger; 60% 36-59 year olds; 16% 60 and older
D) 12% 35 and younger; 58% 36-59 year olds; 30% 60 and older

Question 32

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
$$\begin{array} { r r } x + y + z & = 9 \\2 x - 3 y + 4 z & = 7 \\x - 4 y + 3 z & = - 2\end{array}$$

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) $\left\{ \left( \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } - \frac { 11 } { 5 } , \mathrm { z } \right) \right\}$

Question 33

Apply Gaussian Elimination to Systems with More Variables than Equations
$$\begin{array} { l } x + y + z = 7 \\x - y + 2 z = 7\end{array}$$

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 34

Apply Gaussian Elimination to Systems with More Variables than Equations
$$\begin{array} { r } 3 x + y + z - 2 w = 10 \\2 x + 3 y + 3 z + w = - 5 \\2 x + y + 4 z + 11 w = 11\end{array}$$

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 35

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
$$\begin{array} { r } x + y + z + w = 8 \\3 x + 2 y + z + 4 w = 21 \\4 x + 4 y + 5 z + 8 w = 30 \\2 x + 3 y + 6 z + 9 w = 15\end{array}$$

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 36

Apply Gaussian Elimination to Systems with More Variables than Equations
$$\begin{array} { l } 5 x - y + z = 8 \\7 x + y + z = 6\end{array}$$

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

Question 37

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
$$\begin{array} { r } x - y + z - w = 10 \\- 2 x + 3 y + 5 w = - 28 \\x + 2 y + 8 z + 3 w = - 10 \\x - 4 y - 6 z - 5 w = 30\end{array}$$

A) $\{ ( - 17 w - 10 , - 13 w - 16,5 w + 4 , w ) \}$
B) $\{ ( 3 w - 2 , - 8 w + 3,4 w + 9 , w ) \}$
C) $\{ ( 24,10 , - 6 , - 2 ) \}$
D) $\varnothing$

Question 38

Use Matrices and Gauss-Jordan Elimination to Solve Systems
$$\begin{aligned}3 x + 5 y + 2 w & = - 12 \\2 x + 6 z - w & = - 5 \\- 2 y + 3 z - 3 w & = - 3 \\- x + 2 y + 4 z + w & = - 2\end{aligned}$$

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

Question 39

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
$$\begin{array} { r } x + y + z = 7 \\x - y + 2 z = 7 \\2 x + 3 z = 14\end{array}$$

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 40

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. $$\begin{array} { l | c c c } \mathrm { x } \text { (Number of years after 1980) } & 1 & 5 & 10 \\\hline \mathrm { y } \text { (Number of birds) } & 43 & 139 & 349\end{array}$$ 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) $y = 2 x ^ { 2 } + 12 x + 29$
B) $y = 3 x ^ { 2 } + 24 x + 24$
C) $y = 4 x ^ { 2 } - 12 x + 32$
D) $y = 4 x ^ { 2 } - 36 x + 25$

Question 41

Perform Scalar Multiplication
Let $\mathrm { A } = \left[ \begin{array} { r r } - 3 & 6 \\ 0 & 2 \end{array} \right]$ . Find 4A.

A)
$$\left[ \begin{array} { r r } - 12 & 24 \\0 & 8\end{array} \right]$$
B)
$\left[\begin{array}{rr}-12 & 24 \\0 & 2\end{array}\right]$
C)
$\left[\begin{array}{rr}-12 & 6 \\0 & 2\end{array}\right]$
D)
$\left[\begin{array}{rr}1 & 10 \\4 & 6\end{array}\right]$

Question 42

Add and Subtract Matrices
Let $\mathrm { A } = \left[ \begin{array} { r r } - 1 & 0 \\ 3 & 1 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 1 & 3 \\ 3 & 1 \end{array} \right]$ . Find $\mathrm { A } - \mathrm { B }$ .

A)
$\left[ \begin{array} { r r } 0 & - 3 \\ 0 & 0 \end{array} \right]$
B)
$\left[ \begin{array} { r } - 23 \\ 62 \end{array} \right]$
C)
$\left[ \begin{array} { l l } 0 & 3 \\ 0 & 0 \end{array} \right]$
D)
$[ - 3 ]$

Question 43

Understand What is Meant by Equal Matrices
$$\left[ \begin{array} { c } x \\1\end{array} \right] = \left[ \begin{array} { l } - 1 \\y\end{array} \right]$$

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

Question 44

Understand What is Meant by Equal Matrices
$$\left[ \begin{array} { r r } x + 3 & y + 4 \\7 & 5\end{array} \right] = \left[ \begin{array} { r r } 7 & - 3 \\7 & z\end{array} \right]$$
B) $x = - 4 ; y = 7 ; z = - 5$

A) $x = 4 ; y = - 7 ; z = 5$
D) $x = 4 ; y = 5 ; z = 7$
C) $x = 7 ; y = - 3 ; z = 5$

Question 45

Understand What is Meant by Equal Matrices
$$\left[ \begin{array} { r r } x & y + 6 \\9 z & 3\end{array} \right] = \left[ \begin{array} { c c } 8 & 8 \\90 & 3\end{array} \right]$$

A) $x = 8 ; y = 2 ; z = 10$
B) $x = 8 ; y = 8 ; z = 90$
C) $x = 8 ; y = 3 ; z = 8$
D) $x = 3 ; y = 14 ; z = 810$

Question 46

Matrix Operations and Their Applications
1 Use Matrix Notation
$$\left[ \begin{array} { c c c c c } 3 & 7 & 6 & 7 & - 3 \\3 & - e & - 15 & - 13 & \pi \\- 6 & 9 & - 12 & 12 & 8 \\\frac { 1 } { 3 } & 9 & 6 & - 1 & 5\end{array} \right] ; a _ { 34 }$$

A) $4 \times 5 ; 12$
B) $5 \times 4 ; 6$
C) $20 ; 8$
D) $4 \times 4 ; - 12$

Question 47

Apply Gaussian Elimination to Systems with More Variables than Equations
$$\begin{aligned}2 x + y + 2 z - 4 w & = 10 \\x + 3 y + 2 z - 11 w & = 17 \\3 x + y + 7 z - 21 w & = 0\end{aligned}$$

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 48

Perform Scalar Multiplication
Let $B = \left[ \begin{array} { l l l l } - 1 & 6 & 6 & - 3 \end{array} \right]$ . Find $- 3 B$ .

A) $\left[ \begin{array} { l l l } 3 & - 18 & - 18 \end{array} \right.$ ]
B) $\left[ \begin{array} { l l l } 3 & 6 & 6 \end{array} \right.$ ]
C) $[ - 3 18 18 - 9 ]$
D) $[ - 3 44 - 5 ]$

Question 49

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

A)
$\left[ \begin{array} { l l } 9 & 13 \\ 5 & 18 \end{array} \right]$
B)
$\left[ \begin{array} { l l } 9 & 21 \\ 3 & 30 \end{array} \right]$
C)
$\left[ \begin{array} { l l } 9 & 13 \\ 1 & 10 \end{array} \right]$
D)
$\left[ \begin{array} { c c } 9 & 7 \\ 5 & 10 \end{array} \right]$

Question 50

Add and Subtract Matrices
Let $\mathrm { A } = \left[ \begin{array} { r r r } 2 & 4 & 1 \\ 4 & 2 & - 1 \\ - 1 & 2 & 3 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } 3 & - 4 & 1 \\ - 1 & 0 & 2 \\ 0 & 3 & - 2 \end{array} \right]$ . Find $\mathrm { A } + \mathrm { B }$ .

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

Question 51

Add and Subtract Matrices
Let $\mathrm { A } = \left[ \begin{array} { r r } - 5 & 1 \\ 2 & 5 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 6 & 2 \\ 4 & - 2 \end{array} \right]$ . Find $\mathrm { A } + \mathrm { B }$ .

A)
$\left[\begin{array}{ll}1 & 3 \\6 & 3\end{array}\right]$
B)
$\left[\begin{array}{r}34 \\-13\end{array}\right]$
C)
$\left[\begin{array}{rr}1 & -3 \\-1 & -7\end{array}\right]$

D)
$[ 13 ]$

Question 52

Add and Subtract Matrices
Let $\mathrm { A } = \left[ \begin{array} { r r r } 4 & - 10 & 2 \\ 8 & - 12 & - 3 \\ 4 & 5 & 9 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } 0 & 10 & 3 \\ 4 & 0 & - 5 \\ - 5 & 8 & - 5 \end{array} \right]$ . Find $\mathrm { A } - \mathrm { B }$

A)
$\left[ \begin{array} { r r r } 4 & - 20 & - 1 \\ 4 & - 12 & 2 \\ 9 & - 3 & 14 \end{array} \right]$
B)
$\left[ \begin{array} { r r r } 4 & 0 & - 1 \\ 4 & - 12 & 2 \\ 1 & - 3 & 4 \end{array} \right]$
C)
$\left[ \begin{array} { r r r } 4 & 20 & - 1 \\ 4 & - 12 & 2 \\ 9 & 3 & 14 \end{array} \right]$
D)
$\left[ \begin{array} { r r r } 4 & 0 & 5 \\ 12 & - 12 & - 8 \\ - 1 & 13 & 4 \end{array} \right]$

Question 53

Add and Subtract Matrices
A)
$$\left[ \begin{array} { r r } - 8 & 1 \\- 17 & 0 \\5 & - 6\end{array} \right]$$

A)
$\left[ \begin{array} { r r } -8 & 1 \\ -17 & 8\\ 5 & -6 \end{array} \right]$

B)
$\left[ \begin{array} { r r } 1 & 4 \\ 7 & 8 \\ 11 & 1 \end{array} \right]$
C)
$\left[ \begin{array} { r r } 1 & 1 \\ 7 & 0 \\ 5 & - 2 \end{array} \right]$
D)
$\left[ \begin{array} { r r } 3 & - 2 \\ 7 & 0 \\ - 5 & 6 \end{array} \right]$

Question 54

Solve Problems Involving Systems Without Unique Solutions
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/min between $I _ { 2 }$ and $I _ { 1 } ; t + 3$ cars/min between $I _ { 1 }$ and $I _ { 3 }$
B) $t + 1$ cars/min between $I _ { 2 }$ and $I _ { 1 } ; t + 4$ cars/min between $I _ { 1 }$ and $I _ { 3 }$
C) $t - 2$ cars/min between $I _ { 2 }$ and $I _ { 1 } ; t + 1$ cars/min between $I _ { 1 }$ and $I _ { 3 }$
D) $t + 2$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } - 3$ cars/min between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$

Question 55

Matrix Operations and Their Applications
1 Use Matrix Notation
$$\left[ \begin{array} { r c c c } - 8 & 2 & - 1 & 0 \\5 & - 9 & - 2 & 4\end{array} \right] ; \mathrm { a } _ { 12 }$$

A) $2 \times 4 ; 2$
B) $4 \times 2 ; 2$
C) $2 \times 4 ; 5$
D) $4 \times 2 ; 5$

Question 56

Understand What is Meant by Equal Matrices
$$\left[ \begin{array} { r r } 6 & - 8 \\- 7 & 9\end{array} \right] = \left[ \begin{array} { r r } x y \\- 7 & z\end{array} \right]$$

A) $x = 6 ; y = - 8 ; z = 9$
B) $x = 6 ; y = - 8 ; z = - 7$
C) $x = 6 ; y = - 7 ; z = 9$
D) $x = - 8 ; y = 6 ; z = 9$

Question 57

Solve Problems Involving Systems Without Unique Solutions
The nutritional content per ounce for three foods is given in the table below. $\begin{array}{l|ccc} & \text { Fat }(\mathrm{g} / \mathrm{oz}) & \text { Protein }(\mathrm{g} / \mathrm{oz}) & \text { Fiber }(\mathrm{g} / \mathrm{oz}) \\\hline \text { Food A } & 2 & 4 & 1 \\\text { Food B } & 1 & 2 & 1 \\\text { Food C } & 8 & 16 & 5\end{array}$

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) $3 \mathrm { oz }$ of Food A; $5 \mathrm { oz }$ of Food B; $1 \mathrm { oz }$ of Food $C$
C) 7 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; $2 \mathrm { oz }$ of Food $C$

Question 58

Add and Subtract Matrices
Let $A = \left[ \begin{array} { r r } 5 & - 4 \\ 5 & - 5 \\ - 8 & 7 \end{array} \right]$ and $B = \left[ \begin{array} { r r } - 7 & 6 \\ - 8 & 9 \\ 4 & - 4 \end{array} \right]$ . Find $A + B$

A)
$\left[ \begin{array} { l l l } - 2 & 2 \\ - 3 & 4 \\ - 4 & 3 \end{array} \right]$
B)
$\left[ \begin{array} { r r } 12 & - 10 \\ 13 & - 14 \\ - 12 & 2 \end{array} \right]$
C)
$\left[ \begin{array} { r r } - 2 & 2 \\ 3 & - 5 \\ - 4 & - 3 \end{array} \right]$
D)
$\left[ \begin{array} { r r } - 2 & - 5 \\ - 3 & 4 \\ - 4 & 3 \end{array} \right]$

Question 59

Add and Subtract Matrices
Let $A = \left[ \begin{array} { r } 3 \\ - 5 \\ - 4 \end{array} \right]$ and $B = \left[ \begin{array} { r } - 5 \\ 3 \\ 9 \end{array} \right]$ . Find $A + B$ .

A)
$\left[ \begin{array} { r } - 2 \\ - 2 \\ 5 \end{array} \right]$
B)
$\left[ \begin{array} { l l l } - 2 & - 2 & 5 \end{array} \right]$
C)
$\left[ \begin{array} { r r } 3 & - 5 \\ - 5 & 3 \\ - 4 & 9 \end{array} \right]$
D)
$\left[ \begin{array} { l } 2 \\ 5 \\ 6 \end{array} \right]$

Question 60

Solve Problems Involving Systems Without Unique Solutions
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. $\begin{array}{l|cc} & \begin{array}{l}\text { Hours needed } \\\text { weekly to assemble }\end{array} & \begin{array}{c}\text { Hours needed } \\\text { weekly to test }\end{array} \\\hline \text { Product A } & 1 & 4 \\\text { Product B } & 1 & 5 \\\text { Product C } & 2 & 10\end{array}$

The company has exactly 27 hours per week available for assembly and 120 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) 15 of Product A; $- 2 t + 12$ of Product B
B) 15 t of Product A; $2 t + 12$ of Product $B$
C) $t + 15$ of Product $A ; t + 12$ of Product $B$
D) 15 of Product A; 12 of Product B

Question 61

Solve Matrix Equations
Let $\mathrm { A } = \left[ \begin{array} { r r } 9 & 4 \\ 4 & 9 \\ - 3 & 6 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 2 & - 1 \\ 9 & - 9 \\ - 5 & 4 \end{array} \right] ; \quad \mathrm { X } - \mathrm { B } = \mathrm { A }$

A)
$\left[ \begin{array} { c c } 7 & 3 \\ 13 & 0 \\ - 8 & 10 \end{array} \right]$
B)
$\left[ \begin{array} { r r } 11 & 5 \\ - 5 & 18 \\ 2 & 14 \end{array} \right]$
C)
$\left[ \begin{array} { r r } 7 & 3 \\ - 13 & 9 \\ - 8 & - 10 \end{array} \right]$

D)
$\left[ \begin{array} { r r } 7 & 9 \\ 13 & 0 \\ - 8 & 10 \end{array} \right]$

Question 62

Multiply Matrices
$A = \left[ \begin{array} { r r r } 3 & - 2 & 1 \\ 0 & 4 & - 3 \end{array} \right] , B = \left[ \begin{array} { r r } 3 & 0 \\ - 2 & 1 \end{array} \right]$

A) $\mathrm { AB }$ is not defined.
B)
$\left[\begin{array}{rrr}9 & -6 & 3 \\-6 & 8 & -5\end{array}\right]$

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

Question 63

Solve Matrix Equations
Let $\mathrm { A } = \left[ \begin{array} { c r } 4 & 5 \\ 0 & 3 \\ 9 & - 9 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 9 & - 9 \\ - 4 & 5 \\ 0 & 8 \end{array} \right] ; \quad \mathrm { B } - \mathrm { X } = 3 \mathrm {~A}$

A)
$X = \left[ \begin{array} { c c } - 3 & - 24 \\ - 4 & - 4 \\ - 27 & 35 \end{array} \right]$
B)
$X = \left[ \begin{array} { r r } 21 & 6 \\ 4 & 14 \\ 27 & - 19 \end{array} \right]$
C)
$X = \left[ \begin{array} { r r } - 3 & - 24 \\ 4 & - 4 \\ - 27 & 35 \end{array} \right]$
D)
$X = \left[ \begin{array} { r r } 21 & 6 \\ 4 & 14 \\ 9 & - 19 \end{array} \right]$

Question 64

Multiply Matrices
$$A = \left[ \begin{array} { r r r } 1 & - 6 & 1 \\- 3 & 4 & - 2\end{array} \right] , B = \left[ \begin{array} { l } - 1 \\- 4 \\- 7\end{array} \right]$$

A) $\left[ \begin{array} { r } 16 \\ 1 \end{array} \right]$
B) $\mathrm { AB }$ is not defined.
C) [ $\left. \begin{array} { l l } 16 & 1 \end{array} \right]$
D)
$$\left[ \begin{array} { r c r } 1 & - 6 & 1 \\- 3 & 4 & - 2 \\- 1 & - 4 & - 7\end{array} \right]$$

Question 65

Multiply Matrices
$A = \left[ \begin{array} { r r } - 1 & 3 \\ 1 & 6 \end{array} \right] , B = \left[ \begin{array} { l l l } 0 & - 2 & 5 \\ 1 & - 3 & 2 \end{array} \right]$

A)
$$\left[ \begin{array} { r r r } 3 & - 7 & 1 \\6 & - 20 & 17\end{array} \right]$$
B) $\mathrm { AB }$ is not defined.
C)
$\left[ \begin{array} { r r } 3 & 6 \\ - 7 & - 20 \\ 1 & 17 \end{array} \right]$
D)
$\left[ \begin{array} { c c c } 0 & - 6 & 15 \\ 1 & - 18 & 12 \end{array} \right]$

Question 66

Perform Scalar Multiplication
Let $\mathrm { A } = \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r } - 1 \\ 3 \\ - 2 \end{array} \right]$ . Find $\mathrm { A } - 4 \mathrm {~B}$ .

A)
$\left[ \begin{array} { r } 5 \\ - 15 \\ 10 \end{array} \right]$
B)
$\left[ \begin{array} { r } - 3 \\ 9 \\ - 6 \end{array} \right]$
C)
$\left[ \begin{array} { r } - 5 \\ 15 \\ - 10 \end{array} \right]$
D)
$\left[ \begin{array} { r } 5 \\ - 6 \\ 4 \end{array} \right]$

Question 67

Solve Matrix Equations
Let $A = \left[ \begin{array} { r r } 5 & - 2 \\ - 1 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r } - 1 & - 5 \\ 5 & 5 \end{array} \right] ; \quad X + A = B$

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

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

Question 68

Multiply Matrices
$$A = \left[ \begin{array} { r r r } - 6 & 5 & - 3 \\- 7 & - 8 & - 5 \\- 2 & 2 & - 7\end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 8 & - 1 \\5 & - 6 & 2 \\- 2 & 1 & 1\end{array} \right]$$

A)
$$\left[ \begin{array} { r r r } 25 & 15 & 13 \\- 37 & 99 & - 14 \\22 & - 3 & - 1\end{array} \right]$$
B)
$\left[\begin{array}{rrr}25 & -37 & 22 \\15 & 99 & -3 \\13 & -14 & -1\end{array}\right]$
C)

$$\left[ \begin{array} { r r r } - 6 & 5 & - 3 \\- 7 & - 8 & - 5 \\- 2 & 2 & - 7 \\1 & - 8 & - 1 \\5 & - 6 & 2 \\- 2 & 1 & 1\end{array} \right]$$
D)
$\left[\begin{array}{rrr}-6 & -40 & 3 \\-35 & 48 & -10 \\4 & 2 & -7\end{array}\right]$

Question 69

Solve Matrix Equations
Let $\mathrm { A } = \left[ \begin{array} { r r } 7 & - 7 \\ - 4 & 0 \\ 10 & - 1 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 10 & 0 \\ 0 & - 2 \\ 7 & - 1 \end{array} \right] ; \quad 4 X + A = B$

A)
$\mathrm { X } = \left[ \begin{array} { c c } \frac { 3 } { 4 } & \frac { 7 } { 4 } \\ 1 & - \frac { 1 } { 2 } \\ - \frac { 3 } { 4 } & 0 \end{array} \right] \quad \mathrm { X } = \left[ \begin{array} { c c } - \frac { 3 } { 4 } & - \frac { 7 } { 4 } \\ - 1 & \frac { 1 } { 2 } \\ \frac { 3 } { 4 } & 0 \end{array} \right]$
B)
$X=\left[\begin{array}{cc}-\frac{3}{4} & -\frac{7}{4} \\-1 & \frac{1}{2} \\\frac{3}{4} & 0\end{array}\right]$

C)
$X=\left[\begin{array}{rr}3 & 7 \\4 & -2 \\-3 & 0\end{array}\right]$

D)
$X = \left[ \begin{array} { r r } - 3 & 7 \\ - 4 & 2 \\ 3 & 0 \end{array} \right]$

Question 70

Model Applied Situations with Matrix Operations
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 71

Multiply Matrices
$$A = \left[ \begin{array} { r r } - 1 & 3 \\2 & 2\end{array} \right] , B = \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 5\end{array} \right]$$

A)
$\left[ \begin{array} { l l } - 1 & 15 \\ - 6 & 10 \end{array} \right]$
B)
$\left[ \begin{array} { r r } 2 & 0 \\ - 2 & 10 \end{array} \right]$
C)
$\left[ \begin{array} { r r } 2 & - 6 \\ - 1 & 7 \end{array} \right]$
D)
$\left[ \begin{array} { l l } 15 & - 1 \\ 10 & - 6 \end{array} \right]$

Question 72

Multiply Matrices

A)
$[ - 68 - 79 ]$
B) $\mathrm { AB }$ is not defined.
C)
$\left[ \begin{array} { l } - 68 \\ - 79 \end{array} \right]$
D)

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

Question 73

Multiply Matrices
$A = \left[ \begin{array} { l l l } - 1 & - 9 & 5 \end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 5 & 9 \\ 5 & 6 & 2 \\ - 5 & - 9 & - 8 \end{array} \right]$

A) $[ - 71 - 94 - 67 ]$
B)
$\left[ \begin{array} { l } - 71 \\ - 94 \\ - 67 \end{array} \right]$
C)
$\left[ \begin{array} { r r r } - 1 & - 9 & 5 \\ 1 & - 5 & 9 \\ 5 & 6 & 2 \\ - 5 & - 9 & - 8 \end{array} \right]$
D)
$\left[ \begin{array} { r r r } - 1 & 45 & 45 \\ - 5 & - 54 & 10 \\ 5 & 81 & - 40 \end{array} \right]$

Question 74

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

A) $[ - 34 ]$
B) $[ - 64 ]$
C) $\left[ \begin{array} { l l } - 5 & 4 \end{array} \right]$
D) $\left[ \begin{array} { l l } 0 & 2 \end{array} \right]$

Question 75

Multiply Matrices
$A = \left[ \begin{array} { l l l } - 6 & 2 & 8 \end{array} \right] , B = \left[ \begin{array} { r } 7 \\ 0 \\ - 3 \end{array} \right]$

A) $[ - 66 ]$
B) $[ 330 ]$
C) $[ - 420 - 24 ]$
D)
$$\left[ \begin{array} { r } - 42 \\0 \\- 24\end{array} \right]$$

Question 76

Multiply Matrices
$A = \left[ \begin{array} { r r r } 3 & - 2 & 1 \\ 0 & 4 & - 2 \end{array} \right] , B = \left[ \begin{array} { r r } 5 & 0 \\ - 2 & 2 \end{array} \right]$

A) $\mathrm { AB }$ is not defined.
B) $\left[ \begin{array} { r r r } 15 & - 10 & 5 \\ - 6 & 12 & - 6 \end{array} \right]$
C)
$$\left[ \begin{array} { r r } 15 & - 6 \\- 10 & 12 \\5 & - 6\end{array} \right]$$
D)
$\left[\begin{array}{rr}15 & 0 \\0 & 8\end{array}\right]$

Question 77

Solve Matrix Equations
Let $\mathrm { A } = \left[ \begin{array} { r } 1 - 1 \\ - 3 - 1 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r } - 3 - 4 \\ 1 - 5 \end{array} \right] ; \quad \mathrm { X } + \mathrm { A } = \mathrm { B }$

A)
$X = \left[ \begin{array} { r r } - 4 & - 3 \\ 4 & - 4 \end{array} \right]$
B)
$X = \left[ \begin{array} { r r } - 3 & - 4 \\ - 4 & 4 \end{array} \right]$
C)
$X = \left[ \begin{array} { r r } 4 & - 4 \\- 4 & - 3 \end{array} \right]$
D)
$$X = \left[ \begin{array} { r r } - 4 & 4 \\- 3 & - 4\end{array} \right]$$

Question 78

Solve Matrix Equations
Let $\mathrm { A } = \left[ \begin{array} { r r r } 2 & 1 & - 2 \\ 3 & 0 & 0 \\ 1 & - 2 & 2 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } - 1 & - 2 & - 2 \\ 0 & 1 & 1 \\ 3 & 0 & 2 \end{array} \right] ; \quad 2 \mathrm {~B} - 2 \mathrm {~A} = \mathrm { X }$

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

Question 79

Multiply Matrices
$$A = \left[ \begin{array} { r r r } 1 & 3 & - 1 \\2 & 0 & 3\end{array} \right] , B = \left[ \begin{array} { r r } 3 & 0 \\- 1 & 1 \\0 & 3\end{array} \right]$$

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

Question 80

Perform Scalar Multiplication
Let $A = \left[ \begin{array} { r r r } - 9 & - 3 & 8 \\ - 1 & - 6 & 8 \\ 9 & 6 & 6 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } - 3 & 8 & 9 \\ - 5 & - 4 & 1 \\ 6 & 1 & 9 \end{array} \right]$ . Find 2A - 4B.

A)
$\left[ \begin{array} { r r r } - 6 & - 38 & - 20 \\ 18 & 4 & 12 \\ - 6 & 8 & - 24 \end{array} \right]$
B)
$\left[\begin{array}{rrr}-21 & 2 & 25 \\-7 & -16 & 17 \\24 & 13 & 21\end{array}\right]$
C)
$\left[ \begin{array} { r r r } - 12 & 5 & 17 \\ - 6 & - 10 & 9 \\ 15 & 7 & 15 \end{array} \right]$
D)
$\left[ \begin{array} { r r r } - 12 & - 6 & 15 \\ 5 & - 10 & 7 \\ 17 & 9 & 15 \end{array} \right]$