Question 1

Find the matrix product when possible.
-$\left[ \begin{array} { l l l } - 1 & 2 & 3 \end{array} \right] \left[ \begin{array} { r } 8 \ 0 \ - 3 \end{array} \right]$

Accepted Answer

A) 
[-17] 
B) 
$\left[ \begin{array} { r } - 8 \ 0 \ - 9 \end{array} \right]$ 
C) 
[18] 
D) 
$\left[ \begin{array} { l l l } - 8 & 0 & - 9 \end{array} \right]$

Question 2

Find the value of the determinant.
 $\left| \begin{array} { l l } 
- 10 & 5 \
0 & - 9
\end{array} \right|$

Accepted Answer

A)  90 
B)   -50 
C)   -90 
D)  85

Question 3

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of
the concepts involved in graphing inequalities.
-

Accepted Answer

A)  $y \geq - x$ 
B)  $y \leq x$ 
C)  $y \leq - x$ 
D)  $\mathrm { y } \geq \mathrm { x }$

Question 4

Provide an appropriate response.
-$$\begin{array} { l } 
2 x - 7 y = - 17 \
5 x + 3 y = 19
\end{array}$$ If your friend were going to solve this system of equations by first eliminating y, what general suggestions
would you make so your friend could start on this in a systematic way?

Accepted Answer

The answer of Provide an appropriate response.
-\[\begin{array} { l }...

Question 5

Find the partial fraction decomposition for the rational expression.
$\frac { 2 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 27 } { x ^ { 2 } + x - 2 }$

Accepted Answer

A)  $2 x - 9 + \frac { - 5 / 3 } { x + 2 } + \frac { 1 / 3 } { x - 1 }$ 
B)  $2 x - 5 + \frac { - 7 / 3 } { x + 2 } + \frac { 13 / 3 } { x - 1 }$ 
C)  $2 x - 9 + \frac { - 13 } { x + 2 } + \frac { 7 } { x - 1 }$ 
D)  $2 x - 5 + \frac { - 1 } { x + 2 } + \frac { 8 } { x - 1 }$

Question 6

Use a graphing calculator to find the inverse of the matrix. Give five decimal places, if necessary.
-$$A = \left[ \begin{array} { c c c } 
1.52 & 5.55 & 4 \
- 0.63 & 7.33 & 3.21 \
8.2 & 0.003 & - 2.8
\end{array} \right]$$

Accepted Answer

A) 
$\left[ \begin{array} { l l l } 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{array} \right]$ 
B) 
$$\left[ \begin{array} { r r r } 
0.15171 & - 0.11491 & 0.08500 \
- 0.18145 & 0.27379 & 0.05467 \
0.44411 & - 0.33622 & - 0.10815
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r } 
0.02137 & - 0.01623 & 0.01192 \
- 0.27464 & 0.34436 & 0.00245 \
0.62294 & - 0.47163 & - 0.00792
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r c c } 
0.65789 & 0.18018 & 0.25 \
- 1.58730 & 0.13643 & 0.31153 \
0.12195 & 333.33333 & - 0.35714
\end{array} \right]$$

Question 7

Find the equation of the line y = ax + b that passes through the points (-3, 2) and (2, -8).

Accepted Answer

A)  $y = - 2 x + \frac { 28 } { 5 }$ 
B)  $y = - \frac { 6 } { 5 } x + 1$ 
C)  y= - 2x - 4 
D)  y= - 4x - 2

Question 8

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { l } 
6 x - 7 y = 77 \
4 x - 4 y = 48
\end{array}$$

Accepted Answer

A)  $\{ ( 6 , - 4 ) \}$ 
B)  $\{ ( 7 , - 4 ) \}$ 
C)  Cramer's rule does not apply since $\mathrm { D } = 0 ; \varnothing$ 
D)  $\{ ( 7 , - 5 ) \}$

Question 9

A triangle with vertices at (x1, y1), (x2, y2), and (x3, y3) has area equal to the absolute value of D, where $$D = \frac { 1 } { 2 } \left| \begin{array} { l l l } 
x _ { 1 } & y _ { 1 } & 1 \
x _ { 2 } & y _ { 2 } & 1 \
x _ { 3 } & y _ { 3 } & 1
\end{array} \right|$$ Find the area of the triangle having vertices at P, Q, and R.
-P(0, 5), Q(5, 8), R(7, 4)

Accepted Answer

A)  13 
B)  2 
C)  26 
D)  0

Question 10

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.
$\begin{array} { l } 
x + 3 y - 2 z - w = 10 \
4 x + y + z + 2 w = 10 \
- 3 x - y - 3 z - 2 w = - 7 \
x - y - 3 z - 2 w = - 3
\end{array}$

Accepted Answer

A)  $\{ ( 0,5 , - 5,4 ) \}$ 
B)  $\{ ( 10,10 , - 7 , - 3 ) \}$ 
C)  $\{ ( 1 + w , 3 - 2 w , - 1 + 2 w , w ) \}$ 
D)  $\{ ( 1,3 , - 1,2 ) \}$

Question 11

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

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 12

Write the augmented matrix for the system. Do not
-$$\begin{array} { l } 
6 x + 7 y + 8 z = 63 \
5 x + 3 y + 2 z = 46 \
4 x + 9 y - 2 z = 23
\end{array}$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r | r } 
6 & 5 & 4 & 63 \
7 & 3 & 9 & 46 \
8 & 2 & - 2 & 23
\end{array} \right]$$ 
B) 
$\left[ \begin{array} { r r r | r } 6 & 7 & 8 & 63 \ 5 & 3 & 2 & 46 \ 4 & 9 & - 2 & 23 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r | r } 63 & 8 & 7 & 6 \ 46 & 2 & 3 & 5 \ 23 & - 2 & 9 & 4 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r r } 6 & 7 & 8 \ 5 & 3 & 2 \ 4 & 9 & - 2 \end{array} \right]$

Question 13

A bookstore is having a sale. All books included in the sale have a colored sticker on them to indicate the sale price. There are green stickers, red stickers, and orange stickers. Bob, Sue, and Fred each make purchases of
Books that are on sale. Each row of the table gives information about the numbers of book purchases and the
Total cost of the purchase (before taxes).

Use this information to set up a matrix equation of the form $A X = B$, which can be solved to determine the price each type of sale book. Solve this matrix equation to find the price of a book with a green sticker.
Use the fact that for $A = \left[ \begin{array} { l l l } 1 & 2 & 2 \ 1 & 3 & 2 \ 1 & 2 & 3 \end{array} \right] , A ^ { - 1 } = \left[ \begin{array} { c r r } 5 & - 2 & - 2 \ - 1 & 1 & 0 \ - 1 & 0 & 1 \end{array} \right]$.

Accepted Answer

A)  $4.89 
B)  $4.96 
C)  $4.99 
D)  $4.91

Question 14

Provide an appropriate response.
-Suppose that you are solving a system of three linear equations and have evaluated $D _ { , } D _ { X } , D _ { y }$, and $D _ { Z }$. Under what condition does Cramer's rule not apply?

Accepted Answer

A)  $\mathrm { D } = 0$; and either $\mathrm { D } _ { \mathrm { X } } = 0 , \mathrm { D } _ { \mathrm { y } } = 0$, or $\mathrm { D } _ { \mathrm { Z } } = 0$ 
B)  $\mathrm { D } = 0$ 
C)  $\mathrm { D } _ { \mathrm { X } } = 0$ 
D)  $\mathrm { D } _ { \mathrm { X } } = 0 , \mathrm { D } _ { \mathrm { y } } = 0$, or $\mathrm { D } _ { \mathrm { Z } } = 0$

Question 15

Graph the solution set of the system of inequalities.
-$\begin{array}{r}
2 x+y \geq 4 \
x-1 \geq 0
\end{array}$

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 16

Find the cofactor of the indicated element.
-$a_{11}$
$\left|\begin{array}{rrr}
-1 & 0 & 2 \
3 & 3 & 1 \
4 & 5 & 3
\end{array}\right|$

Accepted Answer

A)  $- 4$ 
B)  4 
C)  12 
D)  $- 3$

Question 17

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
- 2 x + y + 3 z - w = 14 \
- x - 4 y + z - 4 w = 12 \
- 4 x + 3 y - 2 z + w = 4 \
3 x - y - z - 4 w = 3
\end{array}$$

Accepted Answer

A)  $\{ ( - 2,1,2 , - 3 ) \}$ 
B)  $\{ ( - 2 , - 1 , - 2,3 ) \}$ 
C)  $\varnothing$ 
D)  $\{ ( - 4,2,3 , - 5 ) \}$

Question 18

Best Rentals charges a daily fee plus a mileage fee for renting its cars. Barney was charged $90 for 3 days and 300 miles, while Mary was charged $163 for 5 days and 600 miles. What does Best Rental charge per day and
Per mile?

Accepted Answer

A)  $17 per day; 13¢ per mile 
B)  $13 per day; 17¢ per mile 
C)  $16 per day; 14¢ per mile 
D)  $18 per day; 14¢ per mile

Question 19

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

Accepted Answer

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

Question 20

Perform the operation or operations when possible.
-$\left[ \begin{array} { l l l } - 2 & 8 & 4 \end{array} \right] - \left[ \begin{array} { l l } 4 & 3 \end{array} \right]$

Accepted Answer

A)  They cannot be subtracted. 
B)  $\left[ \begin{array} { l l l } - 6 & 8 & 1 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l l } - 6 & 5 & 1 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l l } - 6 & 5 & 4 \end{array} \right]$

Exam 6: Systems and Matrices

Find the matrix product when possible. - $\left[ \begin{array} { l l l } - 1 & 2 & 3 \end{array} \right] \left[ \begin{array} { r } 8 \\ 0 \\ - 3 \end{array} \right]$

Find the value of the determinant. $\left| \begin{array} { l l } - 10 & 5 \\0 & - 9\end{array} \right|$

Provide an appropriate response. - $\begin{array} { l } 2 x - 7 y = - 17 \\5 x + 3 y = 19\end{array}$ If your friend were going to solve this system of equations by first eliminating y, what general suggestions would you make so your friend could start on this in a systematic way?

Find the partial fraction decomposition for the rational expression. $\frac { 2 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 27 } { x ^ { 2 } + x - 2 }$

Use a graphing calculator to find the inverse of the matrix. Give five decimal places, if necessary. - $A = \left[ \begin{array} { c c c } 1.52 & 5.55 & 4 \\- 0.63 & 7.33 & 3.21 \\8.2 & 0.003 & - 2.8\end{array} \right]$

Find the equation of the line y = ax + b that passes through the points (-3, 2) and (2, -8) .

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - $\begin{array} { l } 6 x - 7 y = 77 \\4 x - 4 y = 48\end{array}$

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. $\begin{array} { l } x + 3 y - 2 z - w = 10 \\4 x + y + z + 2 w = 10 \\- 3 x - y - 3 z - 2 w = - 7 \\x - y - 3 z - 2 w = - 3\end{array}$

Write the augmented matrix for the system. Do not - $\begin{array} { l } 6 x + 7 y + 8 z = 63 \\5 x + 3 y + 2 z = 46 \\4 x + 9 y - 2 z = 23\end{array}$

Provide an appropriate response. -Suppose that you are solving a system of three linear equations and have evaluated $D _ { , } D _ { X } , D _ { y }$ , and $D _ { Z }$ . Under what condition does Cramer's rule not apply?

Graph the solution set of the system of inequalities. - $\begin{array}{r}2 x+y \geq 4 \\x-1 \geq 0\end{array}$ $Graph the solution set of the system of inequalities. - \begin{array}{r} 2 x+y \geq 4 \\ x-1 \geq 0 \end{array} A) B) C) D)$

Find the cofactor of the indicated element. - $a_{11}$ $\left|\begin{array}{rrr}-1 & 0 & 2 \\3 & 3 & 1 \\4 & 5 & 3\end{array}\right|$

Solve the system by using the inverse of the coefficient matrix. - $\begin{array} { l } - 2 x + y + 3 z - w = 14 \\- x - 4 y + z - 4 w = 12 \\- 4 x + 3 y - 2 z + w = 4 \\3 x - y - z - 4 w = 3\end{array}$

Best Rentals charges a daily fee plus a mileage fee for renting its cars. Barney was charged $90 for 3 days and 300 miles, while Mary was charged $163 for 5 days and 600 miles. What does Best Rental charge per day and Per mile?

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

Perform the operation or operations when possible. - $\left[ \begin{array} { l l l } - 2 & 8 & 4 \end{array} \right] - \left[ \begin{array} { l l } 4 & 3 \end{array} \right]$

Exam 6: Systems and Matrices

Find the matrix product when possible. - [−123][80−3]\left[ \begin{array} { l l l } - 1 & 2 & 3 \end{array} \right] \left[ \begin{array} { r } 8 \\ 0 \\ - 3 \end{array} \right][−1​2​3​]​80−3​​

Find the value of the determinant. ∣−1050−9∣\left| \begin{array} { l l } - 10 & 5 \\0 & - 9\end{array} \right|​−100​5−9​​

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of the concepts involved in graphing inequalities. -

Find the partial fraction decomposition for the rational expression. 2x3−3x2−2x+27x2+x−2\frac { 2 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 27 } { x ^ { 2 } + x - 2 }x2+x−22x3−3x2−2x+27​

Find the equation of the line y = ax + b that passes through the points (-3, 2) and (2, -8) .

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - 6x−7y=774x−4y=48\begin{array} { l } 6 x - 7 y = 77 \\4 x - 4 y = 48\end{array}6x−7y=774x−4y=48​

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

Write the augmented matrix for the system. Do not - 6x+7y+8z=635x+3y+2z=464x+9y−2z=23\begin{array} { l } 6 x + 7 y + 8 z = 63 \\5 x + 3 y + 2 z = 46 \\4 x + 9 y - 2 z = 23\end{array}6x+7y+8z=635x+3y+2z=464x+9y−2z=23​

Provide an appropriate response. -Suppose that you are solving a system of three linear equations and have evaluated D,DX,DyD _ { , } D _ { X } , D _ { y }D,​DX​,Dy​ , and DZD _ { Z }DZ​ . Under what condition does Cramer's rule not apply?

Graph the solution set of the system of inequalities. - 2x+y≥4x−1≥0\begin{array}{r}2 x+y \geq 4 \\x-1 \geq 0\end{array}2x+y≥4x−1≥0​

Find the cofactor of the indicated element. - a11a_{11}a11​ ∣−102331453∣\left|\begin{array}{rrr}-1 & 0 & 2 \\3 & 3 & 1 \\4 & 5 & 3\end{array}\right|​−134​035​213​​