Question 1

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{array} { l } 
x ^ { 2 } - y ^ { 2 } = 39 \
x - y = 3
\end{array}$$

Accepted Answer

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

Question 2

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { l l l } 
1 & 1 & 1 \
2 & 1 & 1 \
2 & 2 & 3
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { c c c } 
1 & 1 & 1 \
\frac { 1 } { 2 } & 1 & 1 \
\frac { 1 } { 2 } & \frac { 1 } { 2 } & \frac { 1 } { 3 }
\end{array} \right]$$ 
B)  The inverse does not exist. 
C) 
$$\left[ \begin{array} { l l l } 
- 1 & - 1 & - 1 \
- 2 & - 1 & - 1 \
- 2 & - 2 & - 3
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { r r r } 
- 1 & 1 & 0 \
4 & - 1 & - 1 \
- 2 & 0 & 1
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { c c c } 
1 & 1 & 1 \
\frac { 1 } { 2 } & 1 & 1 \
\frac { 1 } { 2 } & \frac { 1 } { 2 } & \frac { 1 } { 3 }
\end{array} \right]$$ 
B)  The inverse does not exist. 
C) 
$$\left[ \begin{array} { l l l } 
- 1 & - 1 & - 1 \
- 2 & - 1 & - 1 \
- 2 & - 2 & - 3
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { r r r } 
- 1 & 1 & 0 \
4 & - 1 & - 1 \
- 2 & 0 & 1
\end{array} \right]$$

Question 3

Provide an appropriate response.
-Let M be a $3 \times 3$ matrix. Which of the following statements are true?
(i) $\mathrm { I } _ { 3 } \mathrm { M } = \mathrm { M }$
(ii) $\mathrm { I } _ { 3 } + \mathrm { M } = \mathrm { M }$
(iii) $\mathrm { MI } _ { 3 } = \mathrm { I } _ { 3 }$
(iv) $\quad \mathrm { MI } _ { 3 } = \mathrm { M }$

Accepted Answer

A)  (i), (iii), (iv) 
B)  (i), (ii), (iii), (iv) 
C)  (ii), (iii) 
D)  (i), (iv) 
A)  (i), (iii), (iv) 
B)  (i), (ii), (iii), (iv) 
C)  (ii), (iii) 
D)  (i), (iv)

Question 4

If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { l } 
\frac { 4 } { x } + \frac { 8 } { y } = 5 \\
\frac { 4 } { x } - \frac { 7 } { y } = - \frac { 5 } { 2 }
\end{array}$$

Accepted Answer

A)  $\left\{ \left\{ \frac { 36 } { 71 } , 2 \right) \right\}$ 
B)  $\{ ( 2,4 ) \}$ 
C)  $\{ ( 4,2 ) \}$ 
D)  $\{ ( - 4 , - 2 ) \}$ 
A)  $\left\{ \left\{ \frac { 36 } { 71 } , 2 \right) \right\}$ 
B)  $\{ ( 2,4 ) \}$ 
C)  $\{ ( 4,2 ) \}$ 
D)  $\{ ( - 4 , - 2 ) \}$

Question 5

Find the matrix product when possible.
-$$\left[ \begin{array} { r r r } 
- 9 & - 5 & - 8 \
4 & - 3 & - 5
\end{array} \right] \left[ \begin{array} { l } 
4 \
8 \
1
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r } 
- 9 & - 5 & - 8 \
4 & - 3 & - 5 \
4 & 8 & 1
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { l } 
- 13 \
- 84
\end{array} \right]$$ 
C) 
$[ - 84 - 13 ]$ 
D) 
$$\left[ \begin{array} { l } 
- 84 \
- 13
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r r } 
- 9 & - 5 & - 8 \
4 & - 3 & - 5 \
4 & 8 & 1
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { l } 
- 13 \
- 84
\end{array} \right]$$ 
C) 
$[ - 84 - 13 ]$ 
D) 
$$\left[ \begin{array} { l } 
- 84 \
- 13
\end{array} \right]$$

Question 6

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth.
-The sum of the squares of the digits of a positive two-digit number is 61 , and the tens digit is 1 more than the units digit. Find the number.

Accepted Answer

A)  56 
B)  45 
C)  54 
D)  65 
A)  56 
B)  45 
C)  54 
D)  65

Question 7

Provide an appropriate response.
-Fill in the blank to complete the statement. Two matrices are equal if they have the same $?$ and if all corresponding elements are equal.

Accepted Answer

A)  dimension 
B)  determinant 
C)  entry 
D)  augmented matrix 
A)  dimension 
B)  determinant 
C)  entry 
D)  augmented matrix

Question 8

Graph the solution set of the system of inequalities.
-$$\begin{array} { c } 
3 x - 2 y \leq 6 \
x - 1 \geq 0
\end{array}$$

Accepted Answer

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

Question 9

Solve the system by elimination.
-$$\begin{array} { l } 
\frac { x + 4 } { 5 } + \frac { 2 y - x } { 10 } = 3 \
\frac { x + 5 } { 4 } + \frac { 3 y + 3 } { 5 } = - 3
\end{array}$$

Accepted Answer

A)  $\left\{ \left\{ 35 , - \frac { 13 } { 2 } \right) \right\}$ 
B)  $\left\{ \left\{ - \frac { 207 } { 2 } , 229 \right) \right\}$ 
C)  $\left\{ \left[ 229 , - \frac { 207 } { 2 } \right) \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ \left\{ 35 , - \frac { 13 } { 2 } \right) \right\}$ 
B)  $\left\{ \left\{ - \frac { 207 } { 2 } , 229 \right) \right\}$ 
C)  $\left\{ \left[ 229 , - \frac { 207 } { 2 } \right) \right\}$ 
D)  $\varnothing$

Question 10

Evaluate the determinant.
-$$\left| \begin{array} { r r r } 
- 1 & - 3 & 0 \
4 & - 3 & 0 \
6 & - 2 & 0
\end{array} \right|$$

Accepted Answer

A)  0 
B)  70 
C)  $- 74$ 
D)  72 
A)  0 
B)  70 
C)  $- 74$ 
D)  72

Question 11

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{array} { l } 
y = x ^ { 2 } - 14 x + 49 \
x + y = 27
\end{array}$$

Accepted Answer

A)  $\{ ( - 2,29 ) , ( - 11,38 ) \}$ 
B)  $\{ ( 2,29 ) , ( 11,16 ) \}$ 
C)  $\{ ( 2,25 ) , ( 11,16 ) \}$ 
D)  $\{ ( 7,20 ) \}$ 
A)  $\{ ( - 2,29 ) , ( - 11,38 ) \}$ 
B)  $\{ ( 2,29 ) , ( 11,16 ) \}$ 
C)  $\{ ( 2,25 ) , ( 11,16 ) \}$ 
D)  $\{ ( 7,20 ) \}$

Question 12

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

Accepted Answer

A) 
$$\left[ \begin{array} { r r } 
- 15 & 5 \
0 & 2
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r } 
- 15 & 25 \
0 & 10
\end{array} \right]$$ 
C)

$\left[ \begin{array} { r r } 2 & 10 \ 5 & 7 \end{array} \right]$ 
D) 
$$\left[ \begin{array} { r r } 
- 15 & 25 \
0 & 2
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r } 
- 15 & 5 \
0 & 2
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r } 
- 15 & 25 \
0 & 10
\end{array} \right]$$ 
C)

$\left[ \begin{array} { r r } 2 & 10 \ 5 & 7 \end{array} \right]$ 
D) 
$$\left[ \begin{array} { r r } 
- 15 & 25 \
0 & 2
\end{array} \right]$$

Question 13

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { r r } 
- 3 & 6 \
1 & - 2
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 16 } & \frac { 3 } { 16 } \
\frac { 1 } { 8 } & \frac { 3 } { 8 }
\end{array} \right]$$ 
B)  The inverse does not exist. 
C) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 8 } & \frac { 3 } { 8 } \
\frac { 1 } { 16 } & \frac { 3 } { 16 }
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { r r } 
\frac { 1 } { 8 } & - \frac { 3 } { 8 } \
- \frac { 1 } { 16 } & \frac { 3 } { 16 }
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 16 } & \frac { 3 } { 16 } \
\frac { 1 } { 8 } & \frac { 3 } { 8 }
\end{array} \right]$$ 
B)  The inverse does not exist. 
C) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 8 } & \frac { 3 } { 8 } \
\frac { 1 } { 16 } & \frac { 3 } { 16 }
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { r r } 
\frac { 1 } { 8 } & - \frac { 3 } { 8 } \
- \frac { 1 } { 16 } & \frac { 3 } { 16 }
\end{array} \right]$$

Question 14

Find the cofactor of the indicated element.
-
$a_{22}$

$$\begin{array} { l } 
\left| \begin{array} { r r r } 
4 & 1 & 1 \
- 2 & - 1 & 1 \
3 & 1 & 4
\end{array} \right|
\end{array}$$

Accepted Answer

A)  $- 1$ 
B)  1 
C)  $- 13$ 
D)  13 
A)  $- 1$ 
B)  1 
C)  $- 13$ 
D)  13

Question 15

Solve the problem using matrices.
-Ellen wishes to mix candy worth $\$ 4.63$ per pound with candy worth $\$ 8.15$ per pound to form 25 pounds of a mixture worth $\$ 7.45$ per pound. How many pounds of the more expensive candy should she use?

Accepted Answer

A)  $5 \mathrm { lb }$ 
B)  $25 \mathrm { lb }$ 
C)  $20 \mathrm { lb }$ 
D)  $7 \mathrm { lb }$ 
A)  $5 \mathrm { lb }$ 
B)  $25 \mathrm { lb }$ 
C)  $20 \mathrm { lb }$ 
D)  $7 \mathrm { lb }$

Question 16

Find the matrix product when possible.
-Given $A = \left[ \begin{array} { l l } 2 & - 3 \ 3 & - 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 0 & - 1 \ - 2 & 3 \end{array} \right]$, find $A B$ and $B A$

Accepted Answer

A) 
$$\mathrm { AB } = \left[ \begin{array} { r r } 
- 3 & 2 \
5 & 0
\end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } 
6 & - 11 \
4 & - 9
\end{array} \right]$$ 
B) 
$$A B = \left[ \begin{array} { r r } 
0 & 3 \
- 6 & - 6
\end{array} \right] ; B A = \left[ \begin{array} { r r } 
0 & 3 \
- 6 & - 6
\end{array} \right]$$ 
C) 
$$A B = \left[ \begin{array} { l l } 
- 6 & 7 \
- 4 & 3
\end{array} \right] ; B A = \left[ \begin{array} { r r } 
3 & - 2 \
- 13 & 12
\end{array} \right]$$ 
D)

$$\mathrm { AB } = \left[ \begin{array} { r r } 
6 & - 11 \
4 & - 9
\end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } 
- 3 & 2 \
5 & 0
\end{array} \right]$$ 
A) 
$$\mathrm { AB } = \left[ \begin{array} { r r } 
- 3 & 2 \
5 & 0
\end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } 
6 & - 11 \
4 & - 9
\end{array} \right]$$ 
B) 
$$A B = \left[ \begin{array} { r r } 
0 & 3 \
- 6 & - 6
\end{array} \right] ; B A = \left[ \begin{array} { r r } 
0 & 3 \
- 6 & - 6
\end{array} \right]$$ 
C) 
$$A B = \left[ \begin{array} { l l } 
- 6 & 7 \
- 4 & 3
\end{array} \right] ; B A = \left[ \begin{array} { r r } 
3 & - 2 \
- 13 & 12
\end{array} \right]$$ 
D)

$$\mathrm { AB } = \left[ \begin{array} { r r } 
6 & - 11 \
4 & - 9
\end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } 
- 3 & 2 \
5 & 0
\end{array} \right]$$

Question 17

Find the matrix product when possible.
-$$\left[ \begin{array} { r r r } 
1 & 3 & - 2 \
2 & 0 & 4
\end{array} \right] \left[ \begin{array} { r r } 
3 & 0 \
- 2 & 1 \
0 & 4
\end{array} \right]$$

Accepted Answer

A)  They cannot be multiplied. 
B) 
$$\left[ \begin{array} { r r } 
- 3 & - 5 \
6 & 16
\end{array} \right]$$ 
C) 
$\left[ \begin{array} { l l } - 5 & - 3 \ 16 & - 6 \end{array} \right]$ 
D) 
$$\left[ \begin{array} { r r r } 
3 & 2 & 0 \
0 & 0 & 16
\end{array} \right]$$ 
A)  They cannot be multiplied. 
B) 
$$\left[ \begin{array} { r r } 
- 3 & - 5 \
6 & 16
\end{array} \right]$$ 
C) 
$\left[ \begin{array} { l l } - 5 & - 3 \ 16 & - 6 \end{array} \right]$ 
D) 
$$\left[ \begin{array} { r r r } 
3 & 2 & 0 \
0 & 0 & 16
\end{array} \right]$$

Question 18

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { c } 
4 x + 4 y - z = 16 \
x - 8 y - 6 z = - 28 \
- 6 x + y + z = - 19
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( - 4,1,8 ) \}$ 
C)  $\{ ( 4,4,1 ) \}$ 
D)  $\{ ( 4,1,4 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( - 4,1,8 ) \}$ 
C)  $\{ ( 4,4,1 ) \}$ 
D)  $\{ ( 4,1,4 ) \}$

Question 19

Perform the operation or operations when possible.
-$$\left[ \begin{array} { r r } 
- 1 & 0 \
3 & 2
\end{array} \right] - \left[ \begin{array} { r r } 
- 1 & 3 \
3 & 1
\end{array} \right]$$

Accepted Answer

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

Question 20

Provide an appropriate response.
-For a certain system of three linear equations, the solution set is $\{ - 17 , - 8,6 \}$, and $D = 8$. Find the values of $D _ { x } , D _ { y }$, and $D _ { z }$

Accepted Answer

A)  $\mathrm { D } _ { \mathrm { X } } = - \frac { 8 } { 17 } ; \mathrm { D } _ { \mathrm { y } } = - 1 ; \mathrm { D } _ { \mathrm { z } } = \frac { 4 } { 3 }$ 
B)  $D _ { x } = - 64 ; D _ { y } = - 136 ; D _ { z } = 48$ 
C)  $D _ { x } = - 17 ; D _ { y } = - 8 ; D _ { z } = 6$ 
D)  $D _ { x } = - 136 ; D _ { y } = - 64 ; D _ { z } = 48$ 
A)  $\mathrm { D } _ { \mathrm { X } } = - \frac { 8 } { 17 } ; \mathrm { D } _ { \mathrm { y } } = - 1 ; \mathrm { D } _ { \mathrm { z } } = \frac { 4 } { 3 }$ 
B)  $D _ { x } = - 64 ; D _ { y } = - 136 ; D _ { z } = 48$ 
C)  $D _ { x } = - 17 ; D _ { y } = - 8 ; D _ { z } = 6$ 
D)  $D _ { x } = - 136 ; D _ { y } = - 64 ; D _ { z } = 48$

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - -=39 x-y=3

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { l l l } 1 & 1 & 1 \\2 & 1 & 1 \\2 & 2 & 3\end{array} \right]$

If the system has infinitely many solutions, write the solution set with x arbitrary. - +=5 -=-

Find the matrix product when possible. - $\left[ \begin{array} { r r r } - 9 & - 5 & - 8 \\4 & - 3 & - 5\end{array} \right] \left[ \begin{array} { l } 4 \\8 \\1\end{array} \right]$

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth. -The sum of the squares of the digits of a positive two-digit number is 61 , and the tens digit is 1 more than the units digit. Find the number.

Provide an appropriate response. -Fill in the blank to complete the statement. Two matrices are equal if they have the same $?$ and if all corresponding elements are equal.

Graph the solution set of the system of inequalities. - 3x-2y\leq6 x-1\geq0 $Graph the solution set of the system of inequalities. - \begin{array} { c } 3 x - 2 y \leq 6 \\ x - 1 \geq 0 \end{array}$

Solve the system by elimination. - +=3 +=-3

Evaluate the determinant. - $\left| \begin{array} { r r r } - 1 & - 3 & 0 \\4 & - 3 & 0 \\6 & - 2 & 0\end{array} \right|$

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - y=-14x+49 x+y=27

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

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { r r } - 3 & 6 \\1 & - 2\end{array} \right]$

Find the cofactor of the indicated element. - $a_{22}$ 4 1 1 -2 -1 1 3 1 4

Solve the problem using matrices. -Ellen wishes to mix candy worth $\$ 4.63$ per pound with candy worth $\$ 8.15$ per pound to form 25 pounds of a mixture worth $\$ 7.45$ per pound. How many pounds of the more expensive candy should she use?

Find the matrix product when possible. -Given $A = \left[ \begin{array} { l l } 2 & - 3 \\ 3 & - 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 0 & - 1 \\ - 2 & 3 \end{array} \right]$ , find $A B$ and $B A$

Find the matrix product when possible. - $\left[ \begin{array} { r r r } 1 & 3 & - 2 \\2 & 0 & 4\end{array} \right] \left[ \begin{array} { r r } 3 & 0 \\- 2 & 1 \\0 & 4\end{array} \right]$

Solve the system by using the inverse of the coefficient matrix. - 4x+4y-z=16 x-8y-6z=-28 -6x+y+z=-19

Perform the operation or operations when possible. - $\left[ \begin{array} { r r } - 1 & 0 \\3 & 2\end{array} \right] - \left[ \begin{array} { r r } - 1 & 3 \\3 & 1\end{array} \right]$

Provide an appropriate response. -For a certain system of three linear equations, the solution set is $\{ - 17 , - 8,6 \}$ , and $D = 8$ . Find the values of $D _ { x } , D _ { y }$ , and $D _ { z }$

Review of Basic Concepts

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

Filters

Exam 10: Systems and Matrices