Question 1

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { r r } 
- 6 & 6 \
- 2 & - 5
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r } 
- \frac { 5 } { 42 } - \frac { 1 } { 7 } \
\frac { 1 } { 21 } - \frac { 1 } { 7 }
\end{array} \right]$$ 
B) 
$\left[\begin{array}{c}
-\frac{1}{7}-\frac{1}{7} \
\frac{1}{21}-\frac{5}{42}
\end{array}\right]$ 
C) 
$\left[\begin{array}{cc}
-\frac{5}{42} & \frac{1}{7} \
-\frac{1}{21} & -\frac{1}{7}
\end{array}\right]$ 
D) 
$\left[\begin{array}{c}
\frac{1}{21}-\frac{1}{7} \
-\frac{5}{42}-\frac{1}{7}
\end{array}\right]$ 
A) 
$$\left[ \begin{array} { r } 
- \frac { 5 } { 42 } - \frac { 1 } { 7 } \
\frac { 1 } { 21 } - \frac { 1 } { 7 }
\end{array} \right]$$ 
B) 
$\left[\begin{array}{c}
-\frac{1}{7}-\frac{1}{7} \
\frac{1}{21}-\frac{5}{42}
\end{array}\right]$ 
C) 
$\left[\begin{array}{cc}
-\frac{5}{42} & \frac{1}{7} \
-\frac{1}{21} & -\frac{1}{7}
\end{array}\right]$ 
D) 
$\left[\begin{array}{c}
\frac{1}{21}-\frac{1}{7} \
-\frac{5}{42}-\frac{1}{7}
\end{array}\right]$

Question 2

Solve the problem.
-The perimeter of a rectangle is 50 cm. The length is 11 cm longer than the width. What are the length and width of the rectangle?

Accepted Answer

A)  Length: 11 cm; width: 7 cm 
B)  Length: 25 cm; width: 14 cm 
C)  Length: 18 cm; width: 7 cm 
D)  Length: 21 cm; width: 10 cm 
A)  Length: 11 cm; width: 7 cm 
B)  Length: 25 cm; width: 14 cm 
C)  Length: 18 cm; width: 7 cm 
D)  Length: 21 cm; width: 10 cm

Question 3

Graph the inequality.
-$x \geq-4$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 4

Find the value of the determinant.
-$\left| \begin{array} { l l l } 5 & 5 & 4 \ 9 & 7 & 3 \ 9 & 4 & 4 \end{array} \right|$

Accepted Answer

A)  911 
B)  $- 73$ 
C)  $- 343$ 
D)  73 
A)  911 
B)  $- 73$ 
C)  $- 343$ 
D)  73

Question 5

Find the dimension of the matrix.
-$$\left[ \begin{array} { l l } 
7 & 0 \
0 & 7
\end{array} \right]$$

Accepted Answer

A)  $2 \times 2$ 
B)  4 
C)  $7 \times 7$ 
D)  7 
A)  $2 \times 2$ 
B)  4 
C)  $7 \times 7$ 
D)  7

Question 6

Graph the solution set of the system of inequalities.
-

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 7

Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities.
-$$y \geq x ^ { 2 } + 5$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 8

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary.
-$5 x - y + z = 8$
 $7 x + y + z = 6$ 
$12 x + 2 z = 14$

Accepted Answer

A)  $\{ ( x , - x - 1 , - 6 x + 7 ) \}$ 
B)  $\{ ( x , - x + 1,6 x - 7 ) \}$ 
C)  $\{ ( x , - x + 1,6 x + 7 ) \}$ 
D)  $\{ ( x , - x + 1 , - 6 x + 7 ) \}$ 
A)  $\{ ( x , - x - 1 , - 6 x + 7 ) \}$ 
B)  $\{ ( x , - x + 1,6 x - 7 ) \}$ 
C)  $\{ ( x , - x + 1,6 x + 7 ) \}$ 
D)  $\{ ( x , - x + 1 , - 6 x + 7 ) \}$

Question 9

Use Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { l l } 
x - y & + 4 z = 3 \
3 x & + z = 0 \
x + 5 y & + z = - 15
\end{array}$$

Accepted Answer

A)  Cramer's rule does not apply since $\mathrm { D } = 0$; $\varnothing$ 
B)  $\{ ( 0,0 , - 3 ) \}$ 
C)  $\{ ( 0 , - 3,3 ) \}$ 
D)  $\{ ( 0 , - 3,0 ) \}$ 
A)  Cramer's rule does not apply since $\mathrm { D } = 0$; $\varnothing$ 
B)  $\{ ( 0,0 , - 3 ) \}$ 
C)  $\{ ( 0 , - 3,3 ) \}$ 
D)  $\{ ( 0 , - 3,0 ) \}$

Question 10

Find the partial fraction decomposition for the rational expression.
-$$\frac { 9 x - 35 } { x ^ { 2 } - 8 x + 15 }$$

Accepted Answer

A)  $\frac { 4 } { x - 5 } + \frac { 5 } { x - 3 }$ 
B)  $\frac { 5 } { x - 5 } + \frac { 4 } { x - 3 }$ 
C)  $\frac { 5 } { x + 5 } + \frac { 4 } { x + 3 }$ 
D)  $\frac { 1 } { x - 5 } + \frac { 1 } { x - 3 }$ 
A)  $\frac { 4 } { x - 5 } + \frac { 5 } { x - 3 }$ 
B)  $\frac { 5 } { x - 5 } + \frac { 4 } { x - 3 }$ 
C)  $\frac { 5 } { x + 5 } + \frac { 4 } { x + 3 }$ 
D)  $\frac { 1 } { x - 5 } + \frac { 1 } { x - 3 }$

Question 11

Graph the inequality.
-$y \leq-1$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 12

Graph the inequality.
-$-2 x-3 y \leq-6$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 13

Solve the linear programming problem.
-An airline with two types of airplanes, $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$, has contracted with a tour group to provide transportation for a minimum of 400 first class, 750 tourist class, and 1500 economy class passengers. For a certain trip, airplane $P _ { 1 }$ costs $\$ 10,000$ to operate and can accommodate 20 first class, 50 tourist class, and 110 economy class passengers. Airplane $P _ { 2 }$ costs $\$ 8500$ to operate and can accommodate 18 first class, 30 tourist class, and 44 economy class passengers. How many of each type of airplane should be used in order to minimize the operating cost?

Accepted Answer

A)  $11 \mathrm { P } _ { 1 }$ planes and $7 \mathrm { P } _ { 2 }$ planes 
B)  $7 \mathrm { P } _ { 1 }$ planes and $11 \mathrm { P } _ { 2 }$ planes 
C)  $5 \mathrm { P } _ { 1 }$ planes and $17 \mathrm { P } _ { 2 }$ planes 
D)  $9 \mathrm { P } _ { 1 }$ planes and $13 \mathrm { P } _ { 2 }$ planes 
A)  $11 \mathrm { P } _ { 1 }$ planes and $7 \mathrm { P } _ { 2 }$ planes 
B)  $7 \mathrm { P } _ { 1 }$ planes and $11 \mathrm { P } _ { 2 }$ planes 
C)  $5 \mathrm { P } _ { 1 }$ planes and $17 \mathrm { P } _ { 2 }$ planes 
D)  $9 \mathrm { P } _ { 1 }$ planes and $13 \mathrm { P } _ { 2 }$ planes

Question 14

Find the matrix product when possible.
-$$\left[ \begin{array} { r r } 
0 & - 1 \
2 & 3
\end{array} \right] \left[ \begin{array} { l l } 
- 2 & 0 \
- 1 & 1
\end{array} \right]$$

Accepted Answer

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

Question 15

Solve the system by substitution.
-$$\begin{array} { l } 
- 2 x = 5 y + 20 \
- 31 = 3 y - 3 x
\end{array}$$

Accepted Answer

A)  $\left\{ \left( - \frac { 122 } { 21 } , \frac { 95 } { 21 } \right) \right\}$ 
B)  $\left\{ \left( - \frac { 95 } { 21 } , - \frac { 122 } { 21 } \right) \right\}$ 
C)  $\left\{ \left( \frac { 95 } { 21 } , - \frac { 122 } { 21 } \right) \right\}$ 
D)  $\left\{ \left( \frac { 85 } { 21 } , - \frac { 122 } { 21 } \right) \right\}$ 
A)  $\left\{ \left( - \frac { 122 } { 21 } , \frac { 95 } { 21 } \right) \right\}$ 
B)  $\left\{ \left( - \frac { 95 } { 21 } , - \frac { 122 } { 21 } \right) \right\}$ 
C)  $\left\{ \left( \frac { 95 } { 21 } , - \frac { 122 } { 21 } \right) \right\}$ 
D)  $\left\{ \left( \frac { 85 } { 21 } , - \frac { 122 } { 21 } \right) \right\}$

Question 16

Provide an appropriate response.
-Suppose that $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ matrix. Under what conditions will the sum $A +$ B exist?

Accepted Answer

A)  If $n = p$ 
B)  If $\mathrm { m } = \mathrm { p }$ and $\mathrm { n } = \mathrm { q }$ 
C)  It is not possible for the sum to exist. 
D)  The sum will always exist. 
A)  If $n = p$ 
B)  If $\mathrm { m } = \mathrm { p }$ and $\mathrm { n } = \mathrm { q }$ 
C)  It is not possible for the sum to exist. 
D)  The sum will always exist.

Question 17

Graph the solution set of the system of inequalities.
-$$\begin{array} { c } 
x - 2 y \leq 2 \
x + y \leq 0
\end{array}$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 18

Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities.
-$$\begin{array} { l } 
y \geq | x + 6 | \
y \leq 3
\end{array}$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 19

Provide an appropriate response.
-Suppose that you are solving a system of three linear equations by the Gauss-Jordan method and obtain the following augmented matrix.
$$\left[ \begin{array} { r r r | r } 
1 & - 2 & 9 & - 8 \
0 & 1 & 8 & - 18 \
0 & 0 & 0 & - 15
\end{array} \right]$$
What conclusion can you draw about the solutions of the system?

Accepted Answer

A)  The system has exactly one solution. 
B)  The system has no solutions. 
C)  The system has infinitely many solutions. 
D)  There is not enough information to draw any conclusion about the solutions of the system. 
A)  The system has exactly one solution. 
B)  The system has no solutions. 
C)  The system has infinitely many solutions. 
D)  There is not enough information to draw any conclusion about the solutions of the system.

Question 20

Find the matrix product when possible.
-$\left[ \begin{array} { r r } 3 & - 2 \ 6 & 0 \end{array} \right] \left[ \begin{array} { r r } 0 & - 2 \ 2 & 6 \end{array} \right]$

Accepted Answer

A) 
$\left[\begin{array}{rr}
-4 & -18 \
0 & -12
\end{array}\right]$ 
B) 
$\left[\begin{array}{rr}
-12 & 0 \
42 & -4
\end{array}\right]$ 
C) 
$\left[\begin{array}{rr}
0 & 4 \
12 & 0
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr}
-18 & -4 \
-12 & 0
\end{array}\right]$ 
A) 
$\left[\begin{array}{rr}
-4 & -18 \
0 & -12
\end{array}\right]$ 
B) 
$\left[\begin{array}{rr}
-12 & 0 \
42 & -4
\end{array}\right]$ 
C) 
$\left[\begin{array}{rr}
0 & 4 \
12 & 0
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr}
-18 & -4 \
-12 & 0
\end{array}\right]$

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { r r } - 6 & 6 \\- 2 & - 5\end{array} \right]$

Solve the problem. -The perimeter of a rectangle is 50 cm. The length is 11 cm longer than the width. What are the length and width of the rectangle?

Graph the inequality. - $x \geq-4$ $Graph the inequality. - x \geq-4$

Find the value of the determinant. - $\left| \begin{array} { l l l } 5 & 5 & 4 \\ 9 & 7 & 3 \\ 9 & 4 & 4 \end{array} \right|$

Find the dimension of the matrix. - $\left[ \begin{array} { l l } 7 & 0 \\0 & 7\end{array} \right]$

Graph the solution set of the system of inequalities. -

Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - $y \geq x ^ { 2 } + 5$ $Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - y \geq x ^ { 2 } + 5$

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. - $5 x - y + z = 8$ $7 x + y + z = 6$ $12 x + 2 z = 14$

Use Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x-y +4z=3 3x +z=0 x+5y +z=-15

Find the partial fraction decomposition for the rational expression. - $\frac { 9 x - 35 } { x ^ { 2 } - 8 x + 15 }$

Graph the inequality. - $y \leq-1$ $Graph the inequality. - y \leq-1$

Graph the inequality. - $-2 x-3 y \leq-6$ $Graph the inequality. - -2 x-3 y \leq-6$

Find the matrix product when possible. - $\left[ \begin{array} { r r } 0 & - 1 \\2 & 3\end{array} \right] \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 1\end{array} \right]$

Solve the system by substitution. - -2x=5y+20 -31=3y-3x

Provide an appropriate response. -Suppose that $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ matrix. Under what conditions will the sum $A +$ B exist?

Graph the solution set of the system of inequalities. - x-2y\leq2 x+y\leq0 $Graph the solution set of the system of inequalities. - \begin{array} { c } x - 2 y \leq 2 \\ x + y \leq 0 \end{array}$

Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - y\geq|x+6| y\leq3 $Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - \begin{array} { l } y \geq | x + 6 | \\ y \leq 3 \end{array}$

Find the matrix product when possible. - $\left[ \begin{array} { r r } 3 & - 2 \\ 6 & 0 \end{array} \right] \left[ \begin{array} { r r } 0 & - 2 \\ 2 & 6 \end{array} \right]$

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Analytic Geometry

Further Topics in Algebra

Review of Basic Concepts

Filters

Exam 9: Systems and Matrices

Find the inverse, if it exists, for the matrix. - [−66−2−5]\left[ \begin{array} { r r } - 6 & 6 \\- 2 & - 5\end{array} \right][−6−2​6−5​]

Solve the problem. -The perimeter of a rectangle is 50 cm. The length is 11 cm longer than the width. What are the length and width of the rectangle?

Graph the inequality. - x≥−4x \geq-4x≥−4

Find the value of the determinant. - ∣554973944∣\left| \begin{array} { l l l } 5 & 5 & 4 \\ 9 & 7 & 3 \\ 9 & 4 & 4 \end{array} \right|​599​574​434​​

Find the dimension of the matrix. - [7007]\left[ \begin{array} { l l } 7 & 0 \\0 & 7\end{array} \right][70​07​]

Graph the solution set of the system of inequalities. -

Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - y≥x2+5y \geq x ^ { 2 } + 5y≥x2+5

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. - 5x−y+z=85 x - y + z = 85x−y+z=8 7x+y+z=67 x + y + z = 67x+y+z=6 12x+2z=1412 x + 2 z = 1412x+2z=14

Use Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x-y +4z=3 3x +z=0 x+5y +z=-15

Find the partial fraction decomposition for the rational expression. - 9x−35x2−8x+15\frac { 9 x - 35 } { x ^ { 2 } - 8 x + 15 }x2−8x+159x−35​

Graph the inequality. - y≤−1y \leq-1y≤−1

Graph the inequality. - −2x−3y≤−6-2 x-3 y \leq-6−2x−3y≤−6

Find the matrix product when possible. - [0−123][−20−11]\left[ \begin{array} { r r } 0 & - 1 \\2 & 3\end{array} \right] \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 1\end{array} \right][02​−13​][−2−1​01​]

Solve the system by substitution. - -2x=5y+20 -31=3y-3x

Provide an appropriate response. -Suppose that AAA is an m×nm \times nm×n matrix and BBB is a p×qp \times qp×q matrix. Under what conditions will the sum A+A +A+ B exist?

Graph the solution set of the system of inequalities. - x-2y\leq2 x+y\leq0

Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - y\geq|x+6| y\leq3

Find the matrix product when possible. - [3−260][0−226]\left[ \begin{array} { r r } 3 & - 2 \\ 6 & 0 \end{array} \right] \left[ \begin{array} { r r } 0 & - 2 \\ 2 & 6 \end{array} \right][36​−20​][02​−26​]

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Analytic Geometry

Further Topics in Algebra

Review of Basic Concepts

Filters

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { r r } - 6 & 6 \\- 2 & - 5\end{array} \right]$

Graph the inequality. - $x \geq-4$ $Graph the inequality. - x \geq-4$

Find the value of the determinant. - $\left| \begin{array} { l l l } 5 & 5 & 4 \\ 9 & 7 & 3 \\ 9 & 4 & 4 \end{array} \right|$

Find the dimension of the matrix. - $\left[ \begin{array} { l l } 7 & 0 \\0 & 7\end{array} \right]$

Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - $y \geq x ^ { 2 } + 5$ $Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - y \geq x ^ { 2 } + 5$

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. - $5 x - y + z = 8$ $7 x + y + z = 6$ $12 x + 2 z = 14$

Find the partial fraction decomposition for the rational expression. - $\frac { 9 x - 35 } { x ^ { 2 } - 8 x + 15 }$

Graph the inequality. - $y \leq-1$ $Graph the inequality. - y \leq-1$

Graph the inequality. - $-2 x-3 y \leq-6$ $Graph the inequality. - -2 x-3 y \leq-6$

Find the matrix product when possible. - $\left[ \begin{array} { r r } 0 & - 1 \\2 & 3\end{array} \right] \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 1\end{array} \right]$

Provide an appropriate response. -Suppose that $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ matrix. Under what conditions will the sum $A +$ B exist?

Graph the solution set of the system of inequalities. - x-2y\leq2 x+y\leq0 $Graph the solution set of the system of inequalities. - \begin{array} { c } x - 2 y \leq 2 \\ x + y \leq 0 \end{array}$

Find the matrix product when possible. - $\left[ \begin{array} { r r } 3 & - 2 \\ 6 & 0 \end{array} \right] \left[ \begin{array} { r r } 0 & - 2 \\ 2 & 6 \end{array} \right]$