Question 1

Find the inverse matrix of A.
-$$A = \left[ \begin{array} { r r r } 
0 & 1 & 1 \
- 5 & 0 & 6 \
0 & 4 & 0
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { c c c } 
\frac { 6 } { 5 } & 0 & 1 \
- \frac { 1 } { 5 } & 0 & 0 \
- \frac { 3 } { 10 } & \frac { 1 } { 4 } & - \frac { 1 } { 4 }
\end{array} \right]$$ 
B) 
$\left[\begin{array}{ccr}
\frac{6}{5} & -\frac{1}{5} & -\frac{3}{10} \
0 & 0 & \frac{1}{4} \
1 & 1 & -\frac{1}{4}
\end{array}\right]$ 
C) 
$\left[\begin{array}{rrr}
\frac{6}{5} & -\frac{1}{5} & -\frac{3}{10} \
0 & 0 & \frac{1}{4} \
1 & 0 & -\frac{1}{4}
\end{array}\right]$ 
D) 
$$\left[ \begin{array} { r r r } 
- \frac { 6 } { 5 } & - \frac { 1 } { 5 } & - \frac { 3 } { 10 } \
- \frac { 1 } { 4 } & 0 & \frac { 1 } { 4 } \
1 & 0 & 0
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { c c c } 
\frac { 6 } { 5 } & 0 & 1 \
- \frac { 1 } { 5 } & 0 & 0 \
- \frac { 3 } { 10 } & \frac { 1 } { 4 } & - \frac { 1 } { 4 }
\end{array} \right]$$ 
B) 
$\left[\begin{array}{ccr}
\frac{6}{5} & -\frac{1}{5} & -\frac{3}{10} \
0 & 0 & \frac{1}{4} \
1 & 1 & -\frac{1}{4}
\end{array}\right]$ 
C) 
$\left[\begin{array}{rrr}
\frac{6}{5} & -\frac{1}{5} & -\frac{3}{10} \
0 & 0 & \frac{1}{4} \
1 & 0 & -\frac{1}{4}
\end{array}\right]$ 
D) 
$$\left[ \begin{array} { r r r } 
- \frac { 6 } { 5 } & - \frac { 1 } { 5 } & - \frac { 3 } { 10 } \
- \frac { 1 } { 4 } & 0 & \frac { 1 } { 4 } \
1 & 0 & 0
\end{array} \right]$$

Question 2

In an analysis of traffic, a certain city estimates the traffic flow as illustrated in the figure below, where the arrows indicate the flow of traffic. If  x1  represents the number of cars traveling from intersection  A  to intersection  B, x2  represents the number of cars traveling from intersection  B  to intersection  C , and so on, we car formulate equations based on the principle that the number of vehicles entering the intersection equals the numt leaving it. Formulate an equation for the traffic at each of the four intersections.

Accepted Answer

A)  
$\begin{array} { l } 
x _ { 1 } - x _ { 4 } = - 650 \
x _ { 1 } - x _ { 2 } = - 450 \
x _ { 2 } - x _ { 3 } = 1400 \
x _ { 3 } - x _ { 4 } = - 1100
\end{array}$ 
B)  
$\begin{array} { l } 
x _ { 1 } - x _ { 4 } = - 450 \
x _ { 1 } - x _ { 2 } = - 650 \
x _ { 2 } - x _ { 3 } = - 1400 \
x _ { 3 } - x _ { 4 } = 1100
\end{array}$ 
C) 
 $\begin{array} { l } 
x _ { 1 } - x _ { 4 } = 450 \
x _ { 1 } - x _ { 2 } = 650 \
x _ { 2 } - x _ { 3 } = 1400 \
x _ { 3 } - x _ { 4 } = - 1100
\end{array}$ 
D)  
$\begin{array} { l } 
x _ { 1 } - x _ { 4 } = 450 \
x _ { 1 } - x _ { 2 } = 650 \
x _ { 2 } - x _ { 3 } = 1400 \
x _ { 3 } - x _ { 4 } = 1100
\end{array}$ 
A)  
$\begin{array} { l } 
x _ { 1 } - x _ { 4 } = - 650 \
x _ { 1 } - x _ { 2 } = - 450 \
x _ { 2 } - x _ { 3 } = 1400 \
x _ { 3 } - x _ { 4 } = - 1100
\end{array}$ 
B)  
$\begin{array} { l } 
x _ { 1 } - x _ { 4 } = - 450 \
x _ { 1 } - x _ { 2 } = - 650 \
x _ { 2 } - x _ { 3 } = - 1400 \
x _ { 3 } - x _ { 4 } = 1100
\end{array}$ 
C) 
 $\begin{array} { l } 
x _ { 1 } - x _ { 4 } = 450 \
x _ { 1 } - x _ { 2 } = 650 \
x _ { 2 } - x _ { 3 } = 1400 \
x _ { 3 } - x _ { 4 } = - 1100
\end{array}$ 
D)  
$\begin{array} { l } 
x _ { 1 } - x _ { 4 } = 450 \
x _ { 1 } - x _ { 2 } = 650 \
x _ { 2 } - x _ { 3 } = 1400 \
x _ { 3 } - x _ { 4 } = 1100
\end{array}$

Question 3

$\left[ \begin{array} { c c c | c } 
1 & 0 & - 3 & 2 \
0 & 1 & 1 & 4 \
0 & 0 & 0 & 0
\end{array} \right]$

Accepted Answer

A)  infinitely many solutions of the form $x = 2 - 3 z , y = 4 - z , z = z$ 
B)  infinitely many solutions of the form $x = 2 + 3 z , y = 4 + z , z = z$ 
C)  infinitely many solutions of the form $x = 2 + 3 z , y = 4 - z , z = z$ 
D)  infinitely many solutions of the form $x = 2 - 3 z , y = 4 + z , z = z$ 
A)  infinitely many solutions of the form $x = 2 - 3 z , y = 4 - z , z = z$ 
B)  infinitely many solutions of the form $x = 2 + 3 z , y = 4 + z , z = z$ 
C)  infinitely many solutions of the form $x = 2 + 3 z , y = 4 - z , z = z$ 
D)  infinitely many solutions of the form $x = 2 - 3 z , y = 4 + z , z = z$

Question 4

Write the augmented matrix associated with the system.
-$$\left\{ \begin{array} { r r } 
- 6 x + 5 y & = 6 \
2 y + & 3 z = - 6 \
- 9 z & = 7
\end{array} \right.$$

Accepted Answer

A) 
$\left[ \begin{array} { r r r | r } - 6 & 5 & 0 & 6 \ 0 & 2 & 3 & - 6 \ 0 & 0 & - 9 & 7 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r | r } - 6 & 5 & 6 & 0 \ 2 & 3 & - 6 & 0 \ - 9 & 7 & 0 & 0 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r | r } - 6 & 5 & 0 & 6 \ 2 & 3 & 0 & - 6 \ 7 & 0 & 0 & 7 \end{array} \right]$ 
D)

$\left[ \begin{array} { r r r | r } - 6 & 5 & 6 & 0 \ 2 & 3 & 0 & - 6 \ - 9 & 0 & 0 & 7 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r r | r } - 6 & 5 & 0 & 6 \ 0 & 2 & 3 & - 6 \ 0 & 0 & - 9 & 7 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r | r } - 6 & 5 & 6 & 0 \ 2 & 3 & - 6 & 0 \ - 9 & 7 & 0 & 0 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r | r } - 6 & 5 & 0 & 6 \ 2 & 3 & 0 & - 6 \ 7 & 0 & 0 & 7 \end{array} \right]$ 
D)

$\left[ \begin{array} { r r r | r } - 6 & 5 & 6 & 0 \ 2 & 3 & 0 & - 6 \ - 9 & 0 & 0 & 7 \end{array} \right]$

Question 5

Solve the matrix equation.
-$$\left[ \begin{array} { c c c } 
1 & - 2 & 3 \
4 & 1 & - 1 \
2 & - 1 & 3
\end{array} \right] \left[ \begin{array} { l } 
\mathrm { x } \
\mathrm { y } \
\mathrm { z }
\end{array} \right] = \left[ \begin{array} { r } 
11 \
4 \
10
\end{array} \right]$$

Accepted Answer

A)  $x = - 2 , y = 3 , z = 5$ 
B)  $x = 2 , y = 3 , z = 5$ 
C)  $x = 2 , y = - 3 , z = 1$ 
D)  $x = - 2 , y = - 3 , z = 1$ 
A)  $x = - 2 , y = 3 , z = 5$ 
B)  $x = 2 , y = 3 , z = 5$ 
C)  $x = 2 , y = - 3 , z = 1$ 
D)  $x = - 2 , y = - 3 , z = 1$

Question 6

Find the inverse matrix of A.
-$$A = \left[ \begin{array} { r r } 
- 3 & 6 \
5 & 0
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { c c } 0 & \frac { 1 } { 5 } \ \frac { 1 } { 6 } & \frac { 1 } { 10 } \end{array} \right]$ 
B)  $\left[ \begin{array} { c c } \frac { 1 } { 6 } & \frac { 1 } { 10 } \ 0 & \frac { 1 } { 5 } \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 0 & - \frac { 1 } { 5 } \ - \frac { 1 } { 6 } & \frac { 1 } { 10 } \end{array} \right]$ 
D)  $\left[ \begin{array} { c c } \frac { 1 } { 10 } & \frac { 1 } { 5 } \ \frac { 1 } { 6 } & 0 \end{array} \right]$ 
A)  $\left[ \begin{array} { c c } 0 & \frac { 1 } { 5 } \ \frac { 1 } { 6 } & \frac { 1 } { 10 } \end{array} \right]$ 
B)  $\left[ \begin{array} { c c } \frac { 1 } { 6 } & \frac { 1 } { 10 } \ 0 & \frac { 1 } { 5 } \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 0 & - \frac { 1 } { 5 } \ - \frac { 1 } { 6 } & \frac { 1 } { 10 } \end{array} \right]$ 
D)  $\left[ \begin{array} { c c } \frac { 1 } { 10 } & \frac { 1 } { 5 } \ \frac { 1 } { 6 } & 0 \end{array} \right]$

Question 7

The sum of a student's three scores is 230. If the first is 19 points more than the second, and the sum of the first two is 23 more than twice the third, then what was the first score?

Accepted Answer

A)  50 
B)  90 
C)  69 
D)  71 
A)  50 
B)  90 
C)  69 
D)  71

Question 8

Decide whether or not matrix A and matrix B are inverses.
-$$A = \left[ \begin{array} { c c } 
10 & 1 \
- 1 & 0
\end{array} \right] \text { and } B = \left[ \begin{array} { r r } 
0 & 1 \
- 1 & 10
\end{array} \right]$$

Accepted Answer

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

Question 9

A bakery sells three types of cakes. The table below gives the number of cups of flour, cups of sugar, and eggs needed to produce each type of cake.  
 To fill its orders for cakes, the bakery used 72 cups of flour, 48 cups of sugar, and 51 eggs. How many cakes of each
Type were made?

Accepted Answer

A)  19 cake I, 8 cake II, 3 cake III 
B)  41 cake I, 8 cake II, 12 cake III 
C)  17 cake I, 8 cake II, 3 cake III 
D)  3 cake I, 8 cake II, 17 cake III 
A)  19 cake I, 8 cake II, 3 cake III 
B)  41 cake I, 8 cake II, 12 cake III 
C)  17 cake I, 8 cake II, 3 cake III 
D)  3 cake I, 8 cake II, 17 cake III

Question 10

Linda invests $25,000 for one year. Part is invested at 5%, another part at 6%, and the rest at 8%. The total income from all 3 investments is $1600. The total income from the 5% and 6% investments is equal to the income
From the 8% investment. Find the amount invested at each rate.

Accepted Answer

A)  $10,000 at 5%, $10,000 at 6%, $5000 at 8% 
B)  $10,000 at 5%, $5000 at 6%, $10,000 at 8% 
C)  $8000 at 5%, $10,000 at 6%, $7000 at 8% 
D)  $5000 at 5%, $10,000 at 6%, $10,000 at 8% 
A)  $10,000 at 5%, $10,000 at 6%, $5000 at 8% 
B)  $10,000 at 5%, $5000 at 6%, $10,000 at 8% 
C)  $8000 at 5%, $10,000 at 6%, $7000 at 8% 
D)  $5000 at 5%, $10,000 at 6%, $10,000 at 8%

Question 11

Find the inverse matrix of A.
-$$A = \left[ \begin{array} { l l l } 
1 & 0 & 8 \
1 & 2 & 3 \
2 & 5 & 3
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { r r r } - 1 & 0 & - 8 \ - 1 & - 2 & - 3 \ - 2 & - 5 & - 3 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l l } 1 & 1 & 2 \ 0 & 2 & 5 \ 8 & 3 & 3 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r r } 9 & - 40 & 16 \ - 3 & 13 & - 5 \ - 1 & 5 & - 2 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } 9 & 40 & 16 \ - 3 & - 13 & - 5 \ - 1 & - 5 & - 2 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r r } - 1 & 0 & - 8 \ - 1 & - 2 & - 3 \ - 2 & - 5 & - 3 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l l } 1 & 1 & 2 \ 0 & 2 & 5 \ 8 & 3 & 3 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r r } 9 & - 40 & 16 \ - 3 & 13 & - 5 \ - 1 & 5 & - 2 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } 9 & 40 & 16 \ - 3 & - 13 & - 5 \ - 1 & - 5 & - 2 \end{array} \right]$

Question 12

$\left[ \begin{array} { r r r | r } 
1 & 0 & 4 & - 2 \
0 & 1 & - 3 & 1 \
0 & 0 & 0 & 0
\end{array} \right]$

Accepted Answer

A)  infinitely many solutions of the form $x = - 2 - 4 z , y = 1 + 3 z , z = z$ 
B)  infinitely many solutions of the form $x = - 2 - 4 z , y = 1 + z , z = z$ 
C)  infinitely many solutions of the form $x = - 2 - z , y = 1 + 3 z , z = z$ 
D)  infinitely many solutions of the form $x = 1 + 3 z , y = - 2 - 4 z , z = z$ 
A)  infinitely many solutions of the form $x = - 2 - 4 z , y = 1 + 3 z , z = z$ 
B)  infinitely many solutions of the form $x = - 2 - 4 z , y = 1 + z , z = z$ 
C)  infinitely many solutions of the form $x = - 2 - z , y = 1 + 3 z , z = z$ 
D)  infinitely many solutions of the form $x = 1 + 3 z , y = - 2 - 4 z , z = z$

Question 13

$\left[ \begin{array} { l l l | l } 
1 & 0 & 5 & 5 \
0 & 1 & 2 & 2 \
0 & 0 & 0 & 1
\end{array} \right]$

Accepted Answer

A)  $x = 0 , y = 0 , z = 1$ 
B)  infinitely many solutions of the form $x = 5 - 5 z , y = 2 - 2 z , z = z$ 
C)  inconsistent system, no solution 
D)  infinitely many solutions of the form $x = 5 z + 5 , y = 2 z + 2 , z = 1$ 
A)  $x = 0 , y = 0 , z = 1$ 
B)  infinitely many solutions of the form $x = 5 - 5 z , y = 2 - 2 z , z = z$ 
C)  inconsistent system, no solution 
D)  infinitely many solutions of the form $x = 5 z + 5 , y = 2 z + 2 , z = 1$

Question 14

A $100,000 trust is to be invested in bonds paying 9%, CDs paying 7%, and mortgages paying 10%. The sum of the bond and CD investment must equal the mortgage investment. To earn an $9100 annual income from the
Investments, how much should the bank invest in bonds?

Accepted Answer

A)  $20,000 
B)  $30,000 
C)  $28,000 
D)  $50,000 
A)  $20,000 
B)  $30,000 
C)  $28,000 
D)  $50,000

Question 15

Use an inverse matrix to find the solution to the system.
-Suppose that in a certain city, the Democratic, Republican, and Independent parties always nominate candidates mayor. It turns out that the percent of people voting for each party candidate in the next election depends on the that voted for that party in the last election. The percent voting for each party in the next election is given by
$$\left[ \begin{array} { l } 
\mathrm { D } \
\mathrm { R } \
\mathrm { I }
\end{array} \right] = \left[ \begin{array} { l l l } 
60 & 20 & 30 \
30 & 60 & 30 \
10 & 20 & 40
\end{array} \right] \left[ \begin{array} { l } 
\mathrm { d } \
\mathrm { r } \
\mathrm { i }
\end{array} \right]$$
where $\mathrm { d } , \mathrm { r }$, and $\mathrm { i }$ are the respective percents (in decimals) that voted for that party in the last election. If the perce voting for a Democrat in the next election is $41 \%$, the percent voting for a Republican in the next election is $42 \%$, and the percent voting for the Independent party in the next election is $17 \%$, then what percent of people voted for an Independent in the last election?

Accepted Answer

A)  40% 
B)  50% 
C)  10% 
D)  25% 
A)  40% 
B)  50% 
C)  10% 
D)  25%

Question 16

Solve using Cramer's rule, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many
solutions.
-$$\left\{ \begin{array} { l } 
3 x + 2 y = 23 \
3 x - 3 y = 3
\end{array} \right.$$

Accepted Answer

A)  $( - 4,5 )$ 
B)  $( 4,5 )$ 
C)  $( - 5 , - 4 )$ 
D)  $( 5,4 )$ 
A)  $( - 4,5 )$ 
B)  $( 4,5 )$ 
C)  $( - 5 , - 4 )$ 
D)  $( 5,4 )$

Question 17

$A = \left[ \begin{array} { r } 
3 \
- 2 \
6
\end{array} \right] \text { and } B = \left[ \begin{array} { r r } 
8 & 7 \
- 3 & - 1 \
7 & 5
\end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { l l } 
72 & 53
\end{array} \right]$ 
B)  not possible 
C)  $\left[ \begin{array} { l } 
72 \
53
\end{array} \right]$ 
D)  $\left[ \begin{array} { r } 
42 \
8 \
72
\end{array} \right]$ 
A)  $\left[ \begin{array} { l l } 
72 & 53
\end{array} \right]$ 
B)  not possible 
C)  $\left[ \begin{array} { l } 
72 \
53
\end{array} \right]$ 
D)  $\left[ \begin{array} { r } 
42 \
8 \
72
\end{array} \right]$

Question 18

Solve the system graphically.
-The sum of the areas of two square plots of land is $2122 \mathrm {~m} ^ { 2 }$ and the difference of their areas is $1240 \mathrm {~m} ^ { 2 }$. Find a length of a side of each plot.

Accepted Answer

A)  $22 \mathrm {~m} , 62 \mathrm {~m}$ 
B)  $41 \mathrm {~m} , 21 \mathrm {~m}$ 
C)  $40 \mathrm {~m} , 20 \mathrm {~m}$ 
D)  $5.25 \mathrm {~m} , 41 \mathrm {~m}$ 
A)  $22 \mathrm {~m} , 62 \mathrm {~m}$ 
B)  $41 \mathrm {~m} , 21 \mathrm {~m}$ 
C)  $40 \mathrm {~m} , 20 \mathrm {~m}$ 
D)  $5.25 \mathrm {~m} , 41 \mathrm {~m}$

Question 19

$\left\{ \begin{array} { l } 
2 x + y - 4 z + w = - 17 \
- x + 2 y + z - 2 w = 7 \
4 x - y + 2 z + 4 w = 11 \
x + y - 2 z - w = - 3
\end{array} \right.$

Accepted Answer

A)   x=4, y=-1, z=5, w=-4 
B)   x=-1, y=3, z=6, w=1 
C)   x=4, y=-2, z=0, w=4 
D)   x=-4, y=-5, z=10, w=4 
A)   x=4, y=-1, z=5, w=-4 
B)   x=-1, y=3, z=6, w=1 
C)   x=4, y=-2, z=0, w=4 
D)   x=-4, y=-5, z=10, w=4

Question 20

$\left\{ \begin{array} { l } 
2 x - 5 z + w = - 1 \
- x + 2 y + z - 2 w = 1 \
4 x - y + 5 w = - 2 \
y - 2 z - w = 0
\end{array} \right.$

Accepted Answer

A)   x=-2, y=4, z=3, w=-5 
B)   x=4, y=-2, z=-7, w=-4 
C)   x=3, y=0, z=-1, w=0 
D)   x=4, y=-2, z=1, w=-4 
A)   x=-2, y=4, z=3, w=-5 
B)   x=4, y=-2, z=-7, w=-4 
C)   x=3, y=0, z=-1, w=0 
D)   x=4, y=-2, z=1, w=-4

Find the inverse matrix of A. - $A = \left[ \begin{array} { r r r } 0 & 1 & 1 \\- 5 & 0 & 6 \\0 & 4 & 0\end{array} \right]$

$\left[ \begin{array} { c c c | c } 1 & 0 & - 3 & 2 \\0 & 1 & 1 & 4 \\0 & 0 & 0 & 0\end{array} \right]$

Write the augmented matrix associated with the system. - $\left\{ \begin{array} { r r } - 6 x + 5 y & = 6 \\2 y + & 3 z = - 6 \\- 9 z & = 7\end{array} \right.$

Solve the matrix equation. - $\left[ \begin{array} { c c c } 1 & - 2 & 3 \\4 & 1 & - 1 \\2 & - 1 & 3\end{array} \right] \left[ \begin{array} { l } \mathrm { x } \\\mathrm { y } \\\mathrm { z }\end{array} \right] = \left[ \begin{array} { r } 11 \\4 \\10\end{array} \right]$

Find the inverse matrix of A. - $A = \left[ \begin{array} { r r } - 3 & 6 \\5 & 0\end{array} \right]$

The sum of a student's three scores is 230. If the first is 19 points more than the second, and the sum of the first two is 23 more than twice the third, then what was the first score?

Decide whether or not matrix A and matrix B are inverses. - $A = \left[ \begin{array} { c c } 10 & 1 \\- 1 & 0\end{array} \right] \text { and } B = \left[ \begin{array} { r r } 0 & 1 \\- 1 & 10\end{array} \right]$

A bakery sells three types of cakes. The table below gives the number of cups of flour, cups of sugar, and eggs needed to produce each type of cake. To fill its orders for cakes, the bakery used 72 cups of flour, 48 cups of sugar, and 51 eggs. How many cakes of each Type were made?

Linda invests $25,000 for one year. Part is invested at 5%, another part at 6%, and the rest at 8%. The total income from all 3 investments is $1600. The total income from the 5% and 6% investments is equal to the income From the 8% investment. Find the amount invested at each rate.

Find the inverse matrix of A. - $A = \left[ \begin{array} { l l l } 1 & 0 & 8 \\1 & 2 & 3 \\2 & 5 & 3\end{array} \right]$

$\left[ \begin{array} { r r r | r } 1 & 0 & 4 & - 2 \\0 & 1 & - 3 & 1 \\0 & 0 & 0 & 0\end{array} \right]$

$\left[ \begin{array} { l l l | l } 1 & 0 & 5 & 5 \\0 & 1 & 2 & 2 \\0 & 0 & 0 & 1\end{array} \right]$

A $100,000 trust is to be invested in bonds paying 9%, CDs paying 7%, and mortgages paying 10%. The sum of the bond and CD investment must equal the mortgage investment. To earn an $9100 annual income from the Investments, how much should the bank invest in bonds?

Solve using Cramer's rule, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many solutions. - $\left\{ \begin{array} { l } 3 x + 2 y = 23 \\3 x - 3 y = 3\end{array} \right.$

$A = \left[ \begin{array} { r } 3 \\- 2 \\6\end{array} \right] \text { and } B = \left[ \begin{array} { r r } 8 & 7 \\- 3 & - 1 \\7 & 5\end{array} \right]$

Solve the system graphically. -The sum of the areas of two square plots of land is $2122 \mathrm {~m} ^ { 2 }$ and the difference of their areas is $1240 \mathrm {~m} ^ { 2 }$ . Find a length of a side of each plot.

$\left\{ \begin{array} { l } 2 x + y - 4 z + w = - 17 \\- x + 2 y + z - 2 w = 7 \\4 x - y + 2 z + 4 w = 11 \\x + y - 2 z - w = - 3\end{array} \right.$

$\left\{ \begin{array} { l } 2 x - 5 z + w = - 1 \\- x + 2 y + z - 2 w = 1 \\4 x - y + 5 w = - 2 \\y - 2 z - w = 0\end{array} \right.$

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Exam 7: Systems of Equations and Matrices

Find the inverse matrix of A. - A=[011−506040]A = \left[ \begin{array} { r r r } 0 & 1 & 1 \\- 5 & 0 & 6 \\0 & 4 & 0\end{array} \right]A=​0−50​104​160​​

[10−3201140000]\left[ \begin{array} { c c c | c } 1 & 0 & - 3 & 2 \\0 & 1 & 1 & 4 \\0 & 0 & 0 & 0\end{array} \right]​100​010​−310​240​​

Write the augmented matrix associated with the system. - {−6x+5y=62y+3z=−6−9z=7\left\{ \begin{array} { r r } - 6 x + 5 y & = 6 \\2 y + & 3 z = - 6 \\- 9 z & = 7\end{array} \right.⎩⎨⎧​−6x+5y2y+−9z​=63z=−6=7​

Find the inverse matrix of A. - A=[−3650]A = \left[ \begin{array} { r r } - 3 & 6 \\5 & 0\end{array} \right]A=[−35​60​]

The sum of a student's three scores is 230. If the first is 19 points more than the second, and the sum of the first two is 23 more than twice the third, then what was the first score?

Decide whether or not matrix A and matrix B are inverses. - A=[101−10] and B=[01−110]A = \left[ \begin{array} { c c } 10 & 1 \\- 1 & 0\end{array} \right] \text { and } B = \left[ \begin{array} { r r } 0 & 1 \\- 1 & 10\end{array} \right]A=[10−1​10​] and B=[0−1​110​]

A bakery sells three types of cakes. The table below gives the number of cups of flour, cups of sugar, and eggs needed to produce each type of cake. To fill its orders for cakes, the bakery used 72 cups of flour, 48 cups of sugar, and 51 eggs. How many cakes of each Type were made?

Linda invests $25,000 for one year. Part is invested at 5%, another part at 6%, and the rest at 8%. The total income from all 3 investments is $1600. The total income from the 5% and 6% investments is equal to the income From the 8% investment. Find the amount invested at each rate.

Find the inverse matrix of A. - A=[108123253]A = \left[ \begin{array} { l l l } 1 & 0 & 8 \\1 & 2 & 3 \\2 & 5 & 3\end{array} \right]A=​112​025​833​​

[104−201−310000]\left[ \begin{array} { r r r | r } 1 & 0 & 4 & - 2 \\0 & 1 & - 3 & 1 \\0 & 0 & 0 & 0\end{array} \right]​100​010​4−30​−210​​

[105501220001]\left[ \begin{array} { l l l | l } 1 & 0 & 5 & 5 \\0 & 1 & 2 & 2 \\0 & 0 & 0 & 1\end{array} \right]​100​010​520​521​​

A $100,000 trust is to be invested in bonds paying 9%, CDs paying 7%, and mortgages paying 10%. The sum of the bond and CD investment must equal the mortgage investment. To earn an $9100 annual income from the Investments, how much should the bank invest in bonds?

Solve using Cramer's rule, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many solutions. - {3x+2y=233x−3y=3\left\{ \begin{array} { l } 3 x + 2 y = 23 \\3 x - 3 y = 3\end{array} \right.{3x+2y=233x−3y=3​

A=[3−26] and B=[87−3−175]A = \left[ \begin{array} { r } 3 \\- 2 \\6\end{array} \right] \text { and } B = \left[ \begin{array} { r r } 8 & 7 \\- 3 & - 1 \\7 & 5\end{array} \right]A=​3−26​​ and B=​8−37​7−15​​

Solve the system graphically. -The sum of the areas of two square plots of land is 2122 m22122 \mathrm {~m} ^ { 2 }2122 m2 and the difference of their areas is 1240 m21240 \mathrm {~m} ^ { 2 }1240 m2 . Find a length of a side of each plot.

{2x+y−4z+w=−17−x+2y+z−2w=74x−y+2z+4w=11x+y−2z−w=−3\left\{ \begin{array} { l } 2 x + y - 4 z + w = - 17 \\- x + 2 y + z - 2 w = 7 \\4 x - y + 2 z + 4 w = 11 \\x + y - 2 z - w = - 3\end{array} \right.⎩⎨⎧​2x+y−4z+w=−17−x+2y+z−2w=74x−y+2z+4w=11x+y−2z−w=−3​

{2x−5z+w=−1−x+2y+z−2w=14x−y+5w=−2y−2z−w=0\left\{ \begin{array} { l } 2 x - 5 z + w = - 1 \\- x + 2 y + z - 2 w = 1 \\4 x - y + 5 w = - 2 \\y - 2 z - w = 0\end{array} \right.⎩⎨⎧​2x−5z+w=−1−x+2y+z−2w=14x−y+5w=−2y−2z−w=0​

Functions, Graphs, and Models; Linear Functions

Linear Models, Equations, and Inequalities

Quadratic, Piecewise-Defined, and Power Functions

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Exponential and Logarithmic Functions

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Find the inverse matrix of A. - $A = \left[ \begin{array} { r r r } 0 & 1 & 1 \\- 5 & 0 & 6 \\0 & 4 & 0\end{array} \right]$

$\left[ \begin{array} { c c c | c } 1 & 0 & - 3 & 2 \\0 & 1 & 1 & 4 \\0 & 0 & 0 & 0\end{array} \right]$

Write the augmented matrix associated with the system. - $\left\{ \begin{array} { r r } - 6 x + 5 y & = 6 \\2 y + & 3 z = - 6 \\- 9 z & = 7\end{array} \right.$

Find the inverse matrix of A. - $A = \left[ \begin{array} { r r } - 3 & 6 \\5 & 0\end{array} \right]$

Decide whether or not matrix A and matrix B are inverses. - $A = \left[ \begin{array} { c c } 10 & 1 \\- 1 & 0\end{array} \right] \text { and } B = \left[ \begin{array} { r r } 0 & 1 \\- 1 & 10\end{array} \right]$

Find the inverse matrix of A. - $A = \left[ \begin{array} { l l l } 1 & 0 & 8 \\1 & 2 & 3 \\2 & 5 & 3\end{array} \right]$

$\left[ \begin{array} { r r r | r } 1 & 0 & 4 & - 2 \\0 & 1 & - 3 & 1 \\0 & 0 & 0 & 0\end{array} \right]$

$\left[ \begin{array} { l l l | l } 1 & 0 & 5 & 5 \\0 & 1 & 2 & 2 \\0 & 0 & 0 & 1\end{array} \right]$

Solve using Cramer's rule, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many solutions. - $\left\{ \begin{array} { l } 3 x + 2 y = 23 \\3 x - 3 y = 3\end{array} \right.$

$A = \left[ \begin{array} { r } 3 \\- 2 \\6\end{array} \right] \text { and } B = \left[ \begin{array} { r r } 8 & 7 \\- 3 & - 1 \\7 & 5\end{array} \right]$

Solve the system graphically. -The sum of the areas of two square plots of land is $2122 \mathrm {~m} ^ { 2 }$ and the difference of their areas is $1240 \mathrm {~m} ^ { 2 }$ . Find a length of a side of each plot.

$\left\{ \begin{array} { l } 2 x + y - 4 z + w = - 17 \\- x + 2 y + z - 2 w = 7 \\4 x - y + 2 z + 4 w = 11 \\x + y - 2 z - w = - 3\end{array} \right.$

$\left\{ \begin{array} { l } 2 x - 5 z + w = - 1 \\- x + 2 y + z - 2 w = 1 \\4 x - y + 5 w = - 2 \\y - 2 z - w = 0\end{array} \right.$