Question 1

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

Accepted Answer

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)  $\{ ( 7 , - 1 , - 6,2 ) \}$ 
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)  $\{ ( 7 , - 1 , - 6,2 ) \}$ A

Question 2

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?

Accepted Answer

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 
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 A

Question 3

Determinants and Cramer's Rule
1 Evaluate a Second-Order Determinant
-$\left| \begin{array} { c c } \frac { 1 } { 3 } & \frac { 1 } { 6 } \ - \frac { 4 } { 7 } & \frac { 3 } { 4 } \end{array} \right|$

Accepted Answer

A)  $\frac { 29 } { 84 }$ 
B)  $\frac { 13 } { 84 }$ 
C)  $\frac { 1 } { 72 }$ 
D)  $- \frac { 53 } { 168 }$ 
A)  $\frac { 29 } { 84 }$ 
B)  $\frac { 13 } { 84 }$ 
C)  $\frac { 1 } { 72 }$ 
D)  $- \frac { 53 } { 168 }$ A

Question 4

Evaluate a Third-Order Determinant
-$$\left| \begin{array} { l l l } 
6 & 3 & 5 \
4 & 1 & 2 \
2 & 3 & 3
\end{array} \right|$$

Accepted Answer

A)  8 
B)  172 
C)  $- 8$ 
D)  $- 16$ ) 
A)  8 
B)  172 
C)  $- 8$ 
D)  $- 16$ )

Question 5

Solve a System of Linear Equations in Three Variables Using Cramer's Rule
-$$\begin{array} { l } 
9 x - 7 y - 3 z = 17 \
4 x + 3 y - 4 z = 45 \
8 x - 8 y - 8 z = - 8
\end{array}$$

Accepted Answer

A)  $\{ ( 8,7,2 ) \}$ 
B)  $\{ ( 7,2,7 ) \}$ 
C)  $\{ ( 8 , - 7 , - 2 ) \}$ 
D)  $\{ ( 9,5,2 ) \}$ 
A)  $\{ ( 8,7,2 ) \}$ 
B)  $\{ ( 7,2,7 ) \}$ 
C)  $\{ ( 8 , - 7 , - 2 ) \}$ 
D)  $\{ ( 9,5,2 ) \}$

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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) 
$$X = \left[ \begin{array} { r r } 
- 4 & 4 \
- 3 & - 4
\end{array} \right]$$ 
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 8

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

Accepted Answer

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

Question 9

Evaluate a Third-Order Determinant
-Determinants are used to show that three points lie on the same line (are collinear). If
$$\left| \begin{array} { l l l } 
x _ { 1 } & y _ { 1 } & 1 \
x _ { 2 } & y _ { 2 } & 1 \
x _ { 3 } & y _ { 3 } & 1
\end{array} \right| = 0 ,$$
then the points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right)$, and $\left( x _ { 3 } , y _ { 3 } \right)$ are collinear. If the determinant does not equal 0 , then the points are not collinear. Are the points $( 4 , - 4 ) , ( 0,3 )$, and $( 20 , - 32 )$ collinear?

Accepted Answer

A)  Yes 
B)  $\mathrm { No }$ 
A)  Yes 
B)  $\mathrm { No }$

Question 10

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

Accepted Answer

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) 
$$\left[ \begin{array} { r r r | r } 
- 4 & 12 & 2 & 45 \
1 & 13 & - 3 & 0 \
2 & - 7 & 4 & 21
\end{array} \right] \quad 
D) 
 \left[ \begin{array} { r r r | r } 
- 4 & 12 & 2 & 9 \
\frac { 1 } { 5 } & \frac { 13 } { 5 } & - \frac { 3 } { 5 } & 0 \
\frac { 2 } { 5 } - \frac { 7 } { 5 } & \frac { 4 } { 5 } & \frac { 21 } { 5 }
\end{array} \right]$$ 
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) 
$$\left[ \begin{array} { r r r | r } 
- 4 & 12 & 2 & 45 \
1 & 13 & - 3 & 0 \
2 & - 7 & 4 & 21
\end{array} \right] \quad 
D) 
 \left[ \begin{array} { r r r | r } 
- 4 & 12 & 2 & 9 \
\frac { 1 } { 5 } & \frac { 13 } { 5 } & - \frac { 3 } { 5 } & 0 \
\frac { 2 } { 5 } - \frac { 7 } { 5 } & \frac { 4 } { 5 } & \frac { 21 } { 5 }
\end{array} \right]$$

Question 11

Use Inverses to Solve Matrix Equations
-$$9 x + 8 y = 86$$
$$4 y = 16$$

Accepted Answer

A) $$\left[ \begin{array} { l l } 
9 & 8 \
0 & 4
\end{array} \right] \left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
86 \
16
\end{array} \right] 
B) 
 \quad \left[ \begin{array} { c c } 
9 & 8 \
4 & 16
\end{array} \right] \left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
86 \
0
\end{array} \right] \quad 
C) 
 \left[ \begin{array} { l l } 
4 & 0 \
9 & 8
\end{array} \right] \left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { r } 
16 \
8
\end{array} \right] 
D) 
 \quad \left[ \begin{array} { l l } 
86 & 8 \
16 & 0
\end{array} \right] \left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
9 \
4
\end{array} \right]$$ 
A) $$\left[ \begin{array} { l l } 
9 & 8 \
0 & 4
\end{array} \right] \left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
86 \
16
\end{array} \right] 
B) 
 \quad \left[ \begin{array} { c c } 
9 & 8 \
4 & 16
\end{array} \right] \left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
86 \
0
\end{array} \right] \quad 
C) 
 \left[ \begin{array} { l l } 
4 & 0 \
9 & 8
\end{array} \right] \left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { r } 
16 \
8
\end{array} \right] 
D) 
 \quad \left[ \begin{array} { l l } 
86 & 8 \
16 & 0
\end{array} \right] \left[ \begin{array} { l } 
x \
y
\end{array} \right] = \left[ \begin{array} { l } 
9 \
4
\end{array} \right]$$

Question 12

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

Accepted Answer

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

Question 13

Use Determinants to Identify Inconsistent Systems and Systems with Dependent Equations
-$$\begin{array} { l } 
4 x - y + 2 z = 1 \
3 x + 5 y - z = 0 \
- 6 x - 10 y + 2 z = 0
\end{array}$$

Accepted Answer

A)  system contains dependent equations 
B)  system is inconsistent 
A)  system contains dependent equations 
B)  system is inconsistent

Question 14

Encode and Decode Messages
-Use the coding matrix $A = \left[ \begin{array} { l l } 3 & 7 \ 2 & 5 \end{array} \right]$ to encode the message LIFE.

Accepted Answer

A)  $\left[ \begin{array} { l l } 99 & 53 \ 69 & 37 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 78 & 62 \ 54 & 43 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 54 & 28 \ 129 & 67 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 3 & - 5 \ 3 & 3 \end{array} \right]$ 
A)  $\left[ \begin{array} { l l } 99 & 53 \ 69 & 37 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 78 & 62 \ 54 & 43 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 54 & 28 \ 129 & 67 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 3 & - 5 \ 3 & 3 \end{array} \right]$

Question 15

Multiplicative Inverses of Matrices and Matrix Equations
1 Find the Multiplicative Inverse of a Square Matrix
-$$A = \left[ \begin{array} { r r r } 
1 & 0 & 0 \
- 5 & 1 & 0 \
0 & 2 & 1
\end{array} \right]$$

Accepted Answer

A) 
$\left[ \begin{array} { r r r } 1 & 0 & 0 \ 5 & 1 & 0 \ - 10 & - 2 & 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r } 1 & 0 & 0 \ - 5 & 1 & 0 \ 0 & 0 & 2 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r } 1 & 2 & 10 \ 0 & 1 & 1 \ 0 & 0 & 1 \end{array} \right]$ 
D)

$\left[ \begin{array} { r r r } 1 & 0 & 0 \ 2 & - 1 & 0 \ 10 & - 5 & 1 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r r } 1 & 0 & 0 \ 5 & 1 & 0 \ - 10 & - 2 & 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r } 1 & 0 & 0 \ - 5 & 1 & 0 \ 0 & 0 & 2 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r } 1 & 2 & 10 \ 0 & 1 & 1 \ 0 & 0 & 1 \end{array} \right]$ 
D)

$\left[ \begin{array} { r r r } 1 & 0 & 0 \ 2 & - 1 & 0 \ 10 & - 5 & 1 \end{array} \right]$

Question 16

Determinants and Cramer's Rule
1 Evaluate a Second-Order Determinant
-$\left| \begin{array} { r r } - 1 & 1 \ 2 & 2 \end{array} \right|$

Accepted Answer

A)  $- 4$ 
B)  0 
C)  4 
D)  $- 5$ 
A)  $- 4$ 
B)  0 
C)  4 
D)  $- 5$

Question 17

Multiplicative Inverses of Matrices and Matrix Equations
1 Find the Multiplicative Inverse of a Square Matrix
-$A = \left[ \begin{array} { r r } - 5 & 6 \ - 2 & - 3 \end{array} \right]$

Accepted Answer

A) 
$$\left[ \begin{array} { c } 
- \frac { 1 } { 9 } - \frac { 2 } { 9 } \
\frac { 2 } { 27 } - \frac { 5 } { 27 }
\end{array} \right]$$ 
B) 
$$
\left[\begin{array}{cc}
-\frac{1}{9} & \frac{2}{9} \
-\frac{2}{27} & -\frac{5}{27}
\end{array}\right]
$$ 
C) 
$\left[\begin{array}{c}
\frac{2}{27}-\frac{5}{27} \
-\frac{1}{9}-\frac{2}{9}
\end{array}\right]$ 
D) 
$$
\left[\begin{array}{c}
-\frac{5}{27}-\frac{2}{9} \
\frac{2}{27}-\frac{1}{9}
\end{array}\right]
$$ 
A) 
$$\left[ \begin{array} { c } 
- \frac { 1 } { 9 } - \frac { 2 } { 9 } \
\frac { 2 } { 27 } - \frac { 5 } { 27 }
\end{array} \right]$$ 
B) 
$$
\left[\begin{array}{cc}
-\frac{1}{9} & \frac{2}{9} \
-\frac{2}{27} & -\frac{5}{27}
\end{array}\right]
$$ 
C) 
$\left[\begin{array}{c}
\frac{2}{27}-\frac{5}{27} \
-\frac{1}{9}-\frac{2}{9}
\end{array}\right]$ 
D) 
$$
\left[\begin{array}{c}
-\frac{5}{27}-\frac{2}{9} \
\frac{2}{27}-\frac{1}{9}
\end{array}\right]
$$

Question 18

Determinants and Cramer's Rule
1 Evaluate a Second-Order Determinant
-$\left| \begin{array} { l l } 6 & 9 \ 9 & 2 \end{array} \right|$

Accepted Answer

A)  $- 69$ 
B)  93 
C)  69 
D)  36 
A)  $- 69$ 
B)  93 
C)  69 
D)  36

Question 19

Multiplicative Inverses of Matrices and Matrix Equations
1 Find the Multiplicative Inverse of a Square Matrix
-$$A = \left[ \begin{array} { r r r } 
2 & - 1 & 0 \
- 1 & 1 & - 2 \
1 & 0 & - 1
\end{array} \right] , \quad B = \left[ \begin{array} { r r r } 
1 & - 1 & 2 \
- 3 & - 2 & 4 \
- 1 & 1 & 1
\end{array} \right]$$

Accepted Answer

A)  $B \neq A ^ { - 1 }$ 
B)  $B = A ^ { - 1 }$ 
A)  $B \neq A ^ { - 1 }$ 
B)  $B = A ^ { - 1 }$

Question 20

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

Accepted Answer

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

Apply Gaussian Elimination to Systems with More Variables than Equations - 3x+y+z-2w=10 2x+3y+3z+w=-5 2x+y+4z+11w=11

Determinants and Cramer's Rule 1 Evaluate a Second-Order Determinant - $\left| \begin{array} { c c } \frac { 1 } { 3 } & \frac { 1 } { 6 } \\ - \frac { 4 } { 7 } & \frac { 3 } { 4 } \end{array} \right|$

Evaluate a Third-Order Determinant - $\left| \begin{array} { l l l } 6 & 3 & 5 \\4 & 1 & 2 \\2 & 3 & 3\end{array} \right|$

Solve a System of Linear Equations in Three Variables Using Cramer's Rule - 9x-7y-3z=17 4x+3y-4z=45 8x-8y-8z=-8

Use Matrices and Gauss-Jordan Elimination to Solve Systems - -3x-y-7z =-76 3x+3y-6z =-27 -9x-2y+z =-23

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

Inconsistent and Dependent Systems and Their Applications 1 Apply Gaussian Elimination to Systems Without Unique Solutions - x+8y+8z =8 7x+7y+z =1 8x+15y+9z =-9

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

Use Inverses to Solve Matrix Equations - $9 x + 8 y = 86$ $4 y = 16$

Inconsistent and Dependent Systems and Their Applications 1 Apply Gaussian Elimination to Systems Without Unique Solutions - 4x-y+3z =12 x+4y+6z =-32 5x+3y+9z =20

Use Determinants to Identify Inconsistent Systems and Systems with Dependent Equations - 4x-y+2z=1 3x+5y-z=0 -6x-10y+2z=0

Encode and Decode Messages -Use the coding matrix $A = \left[ \begin{array} { l l } 3 & 7 \\ 2 & 5 \end{array} \right]$ to encode the message LIFE.

Multiplicative Inverses of Matrices and Matrix Equations 1 Find the Multiplicative Inverse of a Square Matrix - $A = \left[ \begin{array} { r r r } 1 & 0 & 0 \\- 5 & 1 & 0 \\0 & 2 & 1\end{array} \right]$

Determinants and Cramer's Rule 1 Evaluate a Second-Order Determinant - $\left| \begin{array} { r r } - 1 & 1 \\ 2 & 2 \end{array} \right|$

Multiplicative Inverses of Matrices and Matrix Equations 1 Find the Multiplicative Inverse of a Square Matrix - $A = \left[ \begin{array} { r r } - 5 & 6 \\ - 2 & - 3 \end{array} \right]$

Determinants and Cramer's Rule 1 Evaluate a Second-Order Determinant - $\left| \begin{array} { l l } 6 & 9 \\ 9 & 2 \end{array} \right|$

Use Matrices and Gauss-Jordan Elimination to Solve Systems - 3x+5y+2w =-12 2x+6z-w =-5 -2y+3z-3w =-3 -x+2y+4z+w =-2

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Systems of Equations and Inequalities

Conic Sections and Analytic Geometry

Sequences, Induction, and Probability

Prerequisites: Fundamental Concepts of Algebra

Filters

Exam 9: Matrices and Determinants