Question 1

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { r r } 
- 5 & 6 \
0 & 5
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { c c } 
0 & \frac { 1 } { 5 } \
- \frac { 1 } { 5 } & \frac { 6 } { 25 }
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { c c } 
- \frac { 1 } { 5 } & - \frac { 6 } { 25 } \
0 & \frac { 1 } { 5 }
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r } 
- \frac { 1 } { 5 } & \frac { 6 } { 25 } \
0 & \frac { 1 } { 5 }
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 5 } & \frac { 6 } { 25 } \
0 & - \frac { 1 } { 5 }
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { c c } 
0 & \frac { 1 } { 5 } \
- \frac { 1 } { 5 } & \frac { 6 } { 25 }
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { c c } 
- \frac { 1 } { 5 } & - \frac { 6 } { 25 } \
0 & \frac { 1 } { 5 }
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r } 
- \frac { 1 } { 5 } & \frac { 6 } { 25 } \
0 & \frac { 1 } { 5 }
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 5 } & \frac { 6 } { 25 } \
0 & - \frac { 1 } { 5 }
\end{array} \right]$$

Question 2

If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { l } 
3 x + 2 y + z = 4 \
2 x - 3 y - z = 5 \
5 x + 12 y + 5 z = 2
\end{array}$$

Accepted Answer

A)  $\{ ( x , - 5 x - 9 , - 13 x + 22 ) \}$ 
B)  $\{ ( x , 5 x - 9 , - 13 x + 22 ) \}$ 
C)  $\{ ( x , - 5 x + 9 , - 13 x + 22 ) \}$ 
D)  $\{ ( x , 5 x + 9 , - 13 x + 22 ) \}$ 
A)  $\{ ( x , - 5 x - 9 , - 13 x + 22 ) \}$ 
B)  $\{ ( x , 5 x - 9 , - 13 x + 22 ) \}$ 
C)  $\{ ( x , - 5 x + 9 , - 13 x + 22 ) \}$ 
D)  $\{ ( x , 5 x + 9 , - 13 x + 22 ) \}$

Question 3

20Solve the system using a graphing calculator capable of performing row operations. Give solutions with values correct to
the nearest thousandth.
-$$\begin{array} { l } 
0.6 x + 4.9 y - \sqrt { 3 } z = 7 \
\sqrt { 5 } x - 18 y + 10 z = - 1 \
3 x + \sqrt { 2 } y - 4.9 z = \frac { 1 } { 4 }
\end{array}$$

Accepted Answer

A)  $\{ ( - 7.733 , - 5.025,3.491 ) \}$ 
B)  $\{ ( 2.084 , - 1.741,3.529 ) \}$ 
C)  $\{ ( 8.110,0.749,2.619 ) \}$ 
D)  $\{ ( 3.431,1.930,2.606 ) \}$ 
A)  $\{ ( - 7.733 , - 5.025,3.491 ) \}$ 
B)  $\{ ( 2.084 , - 1.741,3.529 ) \}$ 
C)  $\{ ( 8.110,0.749,2.619 ) \}$ 
D)  $\{ ( 3.431,1.930,2.606 ) \}$

Question 4

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { r } 
x + 3 y = - 8 \
- 14 x - 4 y = - 2
\end{array}$$

Accepted Answer

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

Question 5

Find the value of the determinant.
-$$\left| \begin{array} { r r r } 
- 4 & 3 & - 4 \
- 4 & - 2 & - 4 \
4 & 2 & - 2
\end{array} \right|$$

Accepted Answer

A)  $- 24$ 
B)  56 
C)  $- 120$ 
D)  120 
A)  $- 24$ 
B)  56 
C)  $- 120$ 
D)  120

Question 6

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 7

Solve the linear programming problem.
-The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring and 2 man-hours to make one SST ring. How many of each type of ring should be made daily to maximize the company's profit, if the profit on a VIP ring is $\$ 40$ and on an SST ring is $\$ 35$ ?

Accepted Answer

A)  $12 \mathrm { VIP }$ and $12 \mathrm { SST }$ 
B)  18 VIP and 6 SST 
C)  16 VIP and 8 SST 
D)  14 VIP and 10 SST 
A)  $12 \mathrm { VIP }$ and $12 \mathrm { SST }$ 
B)  18 VIP and 6 SST 
C)  16 VIP and 8 SST 
D)  14 VIP and 10 SST

Question 8

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
5 x + 4 y = 8 \
6 x - 3 y = 33
\end{array}$$

Accepted Answer

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

Question 9

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary.
-$\begin{array}{l}
(\log 2) x+(\ln 3) y+(\ln 5) z=4 \
(\ln 3) x+(\log 8) y+(\ln 10) z=11 \
(\log 23) x+(\ln 7) y+(\ln 4) z=15
\end{array}$

Accepted Answer

A)  $\{ ( 4.12321,3.91519,1.27440 ) \}$ 
B)  $\{ ( 0.30103,9.0309,1.38629 ) \}$ 
C)  $\{ ( 9.25474,1.35236 , - 0.16880 ) \}$ 
D)  $\varnothing$ 
A)  $\{ ( 4.12321,3.91519,1.27440 ) \}$ 
B)  $\{ ( 0.30103,9.0309,1.38629 ) \}$ 
C)  $\{ ( 9.25474,1.35236 , - 0.16880 ) \}$ 
D)  $\varnothing$

Question 10

Find the values of the variables for which the statement is true, if possible.
-$\left[ \begin{array} { l l l } 5 & p + 3 & q - 7 \end{array} \right] = \left[ \begin{array} { l l l } k + 2 & 4 & - 6 \end{array} \right]$

Accepted Answer

A)  $\mathrm { k } = - 3 ; \mathrm { p } = - 1 ; \mathrm { q } = - 1$ 
B)  $\mathrm { k } = 3 ; \mathrm { p } = 1 ; \mathrm { q } = 1$ 
C)  $\mathrm { k } = 4 ; \mathrm { p } = 1 ; \mathrm { q } = 0$ 
D)  $\mathrm { k } = 1 ; \mathrm { p } = 5 ; \mathrm { q } = - 6$ 
A)  $\mathrm { k } = - 3 ; \mathrm { p } = - 1 ; \mathrm { q } = - 1$ 
B)  $\mathrm { k } = 3 ; \mathrm { p } = 1 ; \mathrm { q } = 1$ 
C)  $\mathrm { k } = 4 ; \mathrm { p } = 1 ; \mathrm { q } = 0$ 
D)  $\mathrm { k } = 1 ; \mathrm { p } = 5 ; \mathrm { q } = - 6$

Question 11

Solve the system by elimination.
-$$\begin{array} { l } 
\frac { 4 x } { 2 } + \frac { y } { 2 } = - 2 \
\frac { x } { 2 } + \frac { y } { 2 } = 0
\end{array}$$

Accepted Answer

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

Question 12

Find the matrix product when possible.
-Given $A = \left[ \begin{array} { l l } 2 & - 3 \ 3 & - 4 \end{array} \right]$, find $A ^ { 3 }$

Accepted Answer

A) 
$\left[ \begin{array} { r r } 8 & - 27 \ 27 & - 64 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } 80 & - 111 \ 111 & - 154 \end{array} \right]$ 
C)

$\left[ \begin{array} { r r } 8 & - 9 \ 9 & - 10 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } - 28 & - 33 \ 87 & 59 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r } 8 & - 27 \ 27 & - 64 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } 80 & - 111 \ 111 & - 154 \end{array} \right]$ 
C)

$\left[ \begin{array} { r r } 8 & - 9 \ 9 & - 10 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } - 28 & - 33 \ 87 & 59 \end{array} \right]$

Question 13

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last
variable be the arbitrary variable.
-$$\begin{array} { c } 
- 3 x - y - 8 z = - 64 \
4 x + 9 z = 74 \
8 y + z = 14
\end{array}$$

Accepted Answer

A)  $\{ ( 5,6,1 ) \}$ 
B)  $\{ ( - 5,1,10 ) \}$ 
C)  $\{ ( 5,1,6 ) \}$ 
D)  $\varnothing$ 
A)  $\{ ( 5,6,1 ) \}$ 
B)  $\{ ( - 5,1,10 ) \}$ 
C)  $\{ ( 5,1,6 ) \}$ 
D)  $\varnothing$

Question 14

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
2 x + 8 y + 6 z = 20 \
4 x + 2 y - 2 z = - 2
\end{array}$$

Accepted Answer

A)  $\{ ( 2 , - 1,4 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( - 1,2,1 ) \}$ 
D)  $\{ ( 2,1,6 ) \}$ 
A)  $\{ ( 2 , - 1,4 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( - 1,2,1 ) \}$ 
D)  $\{ ( 2,1,6 ) \}$

Question 15

Find the values of the variables for which the statement is true, if possible.
-$$\left[ \begin{array} { r r } 
9 & - 1 \
6 & 7
\end{array} \right] = \left[ \begin{array} { c c } 
x & - 1 \
y & z
\end{array} \right]$$

Accepted Answer

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

Question 16

A nonlinear system is given, along with the graphs of both equations in the system. Determine if the points of
intersection specified on the graph are solutions of the system by substituting directly into both equations.
-$$\begin{array} { l } 
x ^ { 2 } = y - 1 \
y = 3 x + 11
\end{array}$$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 17

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the
solution with y arbitrary.
-$$\begin{array} { l } 
5 x - 2 y - 10 = 0 \
10 x + y - 25 = 0
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\left\{ \left\{ - \frac { 5 } { 2 } y , y \right) \right\}$ 
C)  $\left\{ \left\{ \frac { 12 } { 5 } , 1 \right\} \right\}$ 
D)  $\left\{ \left\{ \frac { 8 } { 5 } , - 1 \right) \right\}$ 
A)  $\varnothing$ 
B)  $\left\{ \left\{ - \frac { 5 } { 2 } y , y \right) \right\}$ 
C)  $\left\{ \left\{ \frac { 12 } { 5 } , 1 \right\} \right\}$ 
D)  $\left\{ \left\{ \frac { 8 } { 5 } , - 1 \right) \right\}$

Question 18

If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { l } 
\frac { 1 } { x } + \frac { 8 } { y } = 3 \\
\frac { 2 } { x } - \frac { 2 } { y } = - 3
\end{array}$$

Accepted Answer

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

Question 19

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants.
-$$\frac { 1 } { a } x + \frac { 1 } { b } y = a b$$
$x + y = a$

Accepted Answer

A)  $\left\{ \left\{ \frac { a ( b - a ) \left( b ^ { 2 } - 1 \right) } { a b ^ { 2 } } , \frac { ( b - a ) ( 1 - a b ) } { a b } \right) \right\}$ 
B)  $\left\{ \left\{ \frac { a ^ { 2 } b \left( b ^ { 2 } - 1 \right) } { b - a } , \frac { a b ( 1 - a b ) } { b - a } \right) \right\}$ 
C)  $\left\{ \left( \frac { a ^ { 2 } \left( b ^ { 2 } - 1 \right) } { b - a } , \frac { a b ( 1 - a b ) } { b - a } \right] \right\}$ 
D)  $\left\{ \left( a b ( 1 - a b ) , a ^ { 2 } \left( b ^ { 2 } - 1 \right) \right) \right\}$ 
A)  $\left\{ \left\{ \frac { a ( b - a ) \left( b ^ { 2 } - 1 \right) } { a b ^ { 2 } } , \frac { ( b - a ) ( 1 - a b ) } { a b } \right) \right\}$ 
B)  $\left\{ \left\{ \frac { a ^ { 2 } b \left( b ^ { 2 } - 1 \right) } { b - a } , \frac { a b ( 1 - a b ) } { b - a } \right) \right\}$ 
C)  $\left\{ \left( \frac { a ^ { 2 } \left( b ^ { 2 } - 1 \right) } { b - a } , \frac { a b ( 1 - a b ) } { b - a } \right] \right\}$ 
D)  $\left\{ \left( a b ( 1 - a b ) , a ^ { 2 } \left( b ^ { 2 } - 1 \right) \right) \right\}$

Question 20

The sizes of two matrices are given. Find the size of the product AB and the size of the product BA, if the given product
can be calculated.
-A is $4 \times 1$; B is $4 \times 1$

Accepted Answer

A)  $\mathrm { AB }$ is $4 \times 4$; BA is $1 \times 1$. 
B)  AB cannot be calculated; BA cannot be calculated. 
C)  $\mathrm { AB }$ is $1 \times 4$; $\mathrm { BA }$ is $4 \times 1$. 
D)  $\mathrm { AB }$ is $4 \times 1$; BA is $1 \times 4$. 
A)  $\mathrm { AB }$ is $4 \times 4$; BA is $1 \times 1$. 
B)  AB cannot be calculated; BA cannot be calculated. 
C)  $\mathrm { AB }$ is $1 \times 4$; $\mathrm { BA }$ is $4 \times 1$. 
D)  $\mathrm { AB }$ is $4 \times 1$; BA is $1 \times 4$.

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { r r } - 5 & 6 \\0 & 5\end{array} \right]$

If the system has infinitely many solutions, write the solution set with x arbitrary. - 3x+2y+z=4 2x-3y-z=5 5x+12y+5z=2

20Solve the system using a graphing calculator capable of performing row operations. Give solutions with values correct to the nearest thousandth. - 0.6x+4.9y-z=7 x-18y+10z=-1 3x+y-4.9z=

Solve the system by using the inverse of the coefficient matrix. - x+3y=-8 -14x-4y=-2

Find the value of the determinant. - $\left| \begin{array} { r r r } - 4 & 3 & - 4 \\- 4 & - 2 & - 4 \\4 & 2 & - 2\end{array} \right|$

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

Solve the system by using the inverse of the coefficient matrix. - 5x+4y=8 6x-3y=33

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary. - (2)x+(3)y+(5)z=4 (3)x+(8)y+(10)z=11 (23)x+(7)y+(4)z=15

Find the values of the variables for which the statement is true, if possible. - $\left[ \begin{array} { l l l } 5 & p + 3 & q - 7 \end{array} \right] = \left[ \begin{array} { l l l } k + 2 & 4 & - 6 \end{array} \right]$

Solve the system by elimination. - +=-2 +=0

Find the matrix product when possible. -Given $A = \left[ \begin{array} { l l } 2 & - 3 \\ 3 & - 4 \end{array} \right]$ , find $A ^ { 3 }$

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable. - -3x-y-8z=-64 4x+9z=74 8y+z=14

Solve the system by using the inverse of the coefficient matrix. - 2x+8y+6z=20 4x+2y-2z=-2

Find the values of the variables for which the statement is true, if possible. - $\left[ \begin{array} { r r } 9 & - 1 \\6 & 7\end{array} \right] = \left[ \begin{array} { c c } x & - 1 \\y & z\end{array} \right]$

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the solution with y arbitrary. - 5x-2y-10=0 10x+y-25=0

If the system has infinitely many solutions, write the solution set with x arbitrary. - +=3 -=-3

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. - $\frac { 1 } { a } x + \frac { 1 } { b } y = a b$ $x + y = a$

The sizes of two matrices are given. Find the size of the product AB and the size of the product BA, if the given product can be calculated. -A is $4 \times 1$ ; B is $4 \times 1$

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

Find the inverse, if it exists, for the matrix. - [−5605]\left[ \begin{array} { r r } - 5 & 6 \\0 & 5\end{array} \right][−50​65​]