Question 1

Find the ratios of products A, B, and C using a closed model.
-

Accepted Answer

A)  $ 6: 3: 3 $ 
B)  $ 1: 3: 3 $ 
C)  $ 2: 3: 3 $ 
D)  $ 3: 2: 3 $ 
A)  $ 6: 3: 3 $ 
B)  $ 1: 3: 3 $ 
C)  $ 2: 3: 3 $ 
D)  $ 3: 2: 3 $

Question 2

Perform the indicated operation, where possible.
-$ \left[\begin{array}{rr}-9 & 1 \ 2 & 5\end{array}\right]+\left[\begin{array}{ll}6 & 2 \ 6 & 3\end{array}\right] $

Accepted Answer

A)  $ \left[\begin{array}{ll}3 & 4 \ 4 & 8\end{array}\right] $ 
B)  $ \left[\begin{array}{rr}-3 & 3 \ 8 & 8\end{array}\right] $ 
C)  $ \left[\begin{array}{ll}-3 & -7 \ -3 & -6\end{array}\right] $ 
D)  Not possible 
A)  $ \left[\begin{array}{ll}3 & 4 \ 4 & 8\end{array}\right] $ 
B)  $ \left[\begin{array}{rr}-3 & 3 \ 8 & 8\end{array}\right] $ 
C)  $ \left[\begin{array}{ll}-3 & -7 \ -3 & -6\end{array}\right] $ 
D)  Not possible

Question 3

Find the ratios of products A, B, and C using a closed model.
-

Accepted Answer

A)  $ 4: 8: 3 $ 
B)  $ 3: 2: 3 $ 
C)  $ 8: 16: 3 $ 
D)  $ 3: 3: 2 $ 
A)  $ 4: 8: 3 $ 
B)  $ 3: 2: 3 $ 
C)  $ 8: 16: 3 $ 
D)  $ 3: 3: 2 $

Question 4

Use the Gauss-Jordan method to solve the system of equations.$$
\begin{array}{l}
-2 x-8 y=6 \
-4 x-16 y=2
\end{array}
$$

Accepted Answer

A)  $ (2,2) $ 
B)  $ \left(-\frac{4}{3} x-\frac{1}{3} y, y\right) $ 
C)  $ (6,2) $ 
D)  No solution 
A)  $ (2,2) $ 
B)  $ \left(-\frac{4}{3} x-\frac{1}{3} y, y\right) $ 
C)  $ (6,2) $ 
D)  No solution

Question 5

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these
products exist.
-A is $ 2 \times 1 $, and $ B $ is $ 1 \times 1 $.

Accepted Answer

A)  $ \mathrm{AB} $ does not exist; $ 1 \times 2 $ 
B)  $ 2 \times 2 ; 1 \times 1 $ 
C)  $ 1 \times 2 ; 1 \times 1 $ 
D)  $ 2 \times 1 $; BA does not exist. 
A)  $ \mathrm{AB} $ does not exist; $ 1 \times 2 $ 
B)  $ 2 \times 2 ; 1 \times 1 $ 
C)  $ 1 \times 2 ; 1 \times 1 $ 
D)  $ 2 \times 1 $; BA does not exist.

Question 6

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these
products exist.
-$ A $ is $ 4 \times 2 $, and B is $ 4 \times 2 $.

Accepted Answer

A)  $ 2 \times 4 ; 4 \times 2 $ 
B)  $ \mathrm{AB} $ does not exist; $ \mathrm{BA} $ does not exist. 
C)  $ 4 \times 4 ; 2 \times 2 $ 
D)  $ 4 \times 2 ; 2 \times 4 $ 
A)  $ 2 \times 4 ; 4 \times 2 $ 
B)  $ \mathrm{AB} $ does not exist; $ \mathrm{BA} $ does not exist. 
C)  $ 4 \times 4 ; 2 \times 2 $ 
D)  $ 4 \times 2 ; 2 \times 4 $

Question 7

Does a matrix with a column of all zeros have an inverse? Why or why not?

Accepted Answer

No. Assume that the matrix had an invers

Question 8

Use the indicated row operation to change the matrix.
-Replace $ \mathrm{R}_{2} $ by $ \mathrm{R}_{1}+(-1) \mathrm{R}_{2} $.
$ \left[\begin{array}{rr|r}1 & -3 & 4 \ 2 & 3 & 1\end{array}\right] $

Accepted Answer

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

Question 9

Solve the problem.
-A company makes three chocolate candies: cherry, almond, and raisin. Matrix A gives the amount of ingredients in one batch. Matrix B gives the costs of ingredients from suppliers J and K. What is The cost of 100 batches of each candy using ingredients from supplier K?

Accepted Answer

A)  $ \$ 12.000 $ 
B)  $ \$ 15.200 $ 
C)  $ \$ 6000 $ 
D)  $ \$ 9400 $ 
A)  $ \$ 12.000 $ 
B)  $ \$ 15.200 $ 
C)  $ \$ 6000 $ 
D)  $ \$ 9400 $

Question 10

Find the matrix product, if possible.
-$
\left[\begin{array}{rrr}
1 & 3 & -1 \
3 & 0 & 5
\end{array}\right]\left[\begin{array}{rr}
3 & 0 \
-1 & 1 \
0 & 5
\end{array}\right]
$

Accepted Answer

A)  $ \left[\begin{array}{cc}-2 & 0 \ 25 & 9\end{array}\right] $ 
B)  $ \left[\begin{array}{rrr}3 & -3 & 0 \ 0 & 0 & 25\end{array}\right] $ 
C)  $ \left[\begin{array}{ll}0 & -2 \ 9 & 25\end{array}\right] $ 
D)  Does not exist 
A)  $ \left[\begin{array}{cc}-2 & 0 \ 25 & 9\end{array}\right] $ 
B)  $ \left[\begin{array}{rrr}3 & -3 & 0 \ 0 & 0 & 25\end{array}\right] $ 
C)  $ \left[\begin{array}{ll}0 & -2 \ 9 & 25\end{array}\right] $ 
D)  Does not exist

Question 11

Solve the problem.
-A recording company is to release 60 new CDs in the categories of rock, country, jazz, and classical. If twice the number of rock CDs is to equal three times the number of country CDs and if the number of jazz CDs is to equal the number of classical CDs, how can the CDs be distributed among the four types? (Let $ x $ be the number of rock CDs, $ y $ the number of country CDs, $ z $ the number of jazz CDs, and $ w $ the number of classical CDs. Let $ w $ be the parameter.)

Accepted Answer

A)  $ x=36-w, y=16-w, z=w, 0 \leq w \leq 20 $ 
B)  $ x=30-3 w / 5, y=20-2 w / 5, z=w, 0 \leq w \leq 26 $ 
C)  $ x=36-6 w / 5, y=24-4 w / 5, z=w, 0 \leq w \leq 30 $ 
D)  $ x=30-2 w, y=30-2 w / 5, z=w, 0 \leq w \leq 30 $ 
A)  $ x=36-w, y=16-w, z=w, 0 \leq w \leq 20 $ 
B)  $ x=30-3 w / 5, y=20-2 w / 5, z=w, 0 \leq w \leq 26 $ 
C)  $ x=36-6 w / 5, y=24-4 w / 5, z=w, 0 \leq w \leq 30 $ 
D)  $ x=30-2 w, y=30-2 w / 5, z=w, 0 \leq w \leq 30 $

Question 12

Write a matrix to display the information.
-Four fire towers are linked by communication devices as shown in the diagram. Assume that each tower cannot communicate with itself and that communication with the other stations takes place Only in the direction of the arrows. Write a 4 × 4 matrix in which rows (and columns)1, 2, 3, and 4 Refer to W, X, Y, and Z, respectively. Make an element a 1 if the tower for that row communicates With the tower for that column, and otherwise make the element a 0.

Accepted Answer

A)  $ \left[\begin{array}{llll}0 & 0 & 1 & 0 \ 0 & 1 & 0 & 1 \ 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 0\end{array}\right] \quad $ 
B)  $ \left[\begin{array}{llll}0 & 1 & 1 & 0 \ 1 & 0 & 1 & 0 \ 1 & 1 & 0 & 1 \ 0 & 0 & 1 & 0\end{array}\right] \quad $ 
C)  $ \left[\begin{array}{llll}1 & 0 & 1 & 1 \ 0 & 1 & 0 & 1 \ 1 & 0 & 1 & 0 \ 1 & 1 & 0 & 1\end{array}\right] \quad $ 
D)  $ \left[\begin{array}{llll}0 & 1 & 0 & 0 \ 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 1 \ 0 & 0 & 1 & 0\end{array}\right] $ 
A)  $ \left[\begin{array}{llll}0 & 0 & 1 & 0 \ 0 & 1 & 0 & 1 \ 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 0\end{array}\right] \quad $ 
B)  $ \left[\begin{array}{llll}0 & 1 & 1 & 0 \ 1 & 0 & 1 & 0 \ 1 & 1 & 0 & 1 \ 0 & 0 & 1 & 0\end{array}\right] \quad $ 
C)  $ \left[\begin{array}{llll}1 & 0 & 1 & 1 \ 0 & 1 & 0 & 1 \ 1 & 0 & 1 & 0 \ 1 & 1 & 0 & 1\end{array}\right] \quad $ 
D)  $ \left[\begin{array}{llll}0 & 1 & 0 & 0 \ 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 1 \ 0 & 0 & 1 & 0\end{array}\right] $

Question 13

Find the ratios of products A, B, and C using a closed model.
-

Accepted Answer

A)  32: 40: 13 
B)  $ 8: 10: 3 $ 
C)  $ 8: 10: 40 $ 
D)  $ 8: 13: 32 $ 
A)  32: 40: 13 
B)  $ 8: 10: 3 $ 
C)  $ 8: 10: 40 $ 
D)  $ 8: 13: 32 $

Question 14

Find the values of the variables in the equation.
-$$
\left[\begin{array}{crr}
t-1 & -4 & 0 \
-3 & 4 & 3
\end{array}\right]=\left[\begin{array}{lll}
-4 & -4 & 0 \
-3 & x-8 & 3
\end{array}\right]
$$

Accepted Answer

A)  $ t=-3, x=12 $ 
B)  $ t=-4, x=4 $ 
C)  $ \mathrm{t}=-3, \mathrm{x}=-4 $ 
D)  $ t=-4, x=-4 $ 
A)  $ t=-3, x=12 $ 
B)  $ t=-4, x=4 $ 
C)  $ \mathrm{t}=-3, \mathrm{x}=-4 $ 
D)  $ t=-4, x=-4 $

Question 15

Use the echelon method to solve the system.$$
\begin{array}{l}
\frac{x}{5}+\frac{y}{5}=1 \
\frac{x}{5}-\frac{y}{5}=-\frac{7}{5}
\end{array}
$$

Accepted Answer

A)  $ (1,7) $ 
B)  $ (-2,7) $ 
C)  $ (-1,6) $ 
D)  No solution 
A)  $ (1,7) $ 
B)  $ (-2,7) $ 
C)  $ (-1,6) $ 
D)  No solution

Question 16

Perform the indicated operation, where possible.
-$ \left[\begin{array}{rr}-1 & 5 \ 0 & 4 \ 8 & -4\end{array}\right]-\left[\begin{array}{ll}2 & 1 \ 7 & 4 \ 2 & 2\end{array}\right] $

Accepted Answer

A)  $ \left[\begin{array}{cc}3 & -4 \ 7 & 0 \ -6 & 6\end{array}\right] $ 
B)  $ \left[\begin{array}{rr}-3 & 4 \ -7 & 0 \ 6 & -6\end{array}\right] $ 
C)  $ \left[\begin{array}{rr}1 & 6 \ 7 & 8 \ 10 & -2\end{array}\right] $ 
D)  Not possible 
A)  $ \left[\begin{array}{cc}3 & -4 \ 7 & 0 \ -6 & 6\end{array}\right] $ 
B)  $ \left[\begin{array}{rr}-3 & 4 \ -7 & 0 \ 6 & -6\end{array}\right] $ 
C)  $ \left[\begin{array}{rr}1 & 6 \ 7 & 8 \ 10 & -2\end{array}\right] $ 
D)  Not possible

Question 17

Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.)
-$$
\begin{aligned}
1.5 x+3.1 y-2.3 z+0.6 w &=2.7 \
6.1 x-0.5 z-3.4 w &=0.8 \
11.5 y-2.2 w &=0 \
4.0 x+3.0 y-2.1 z &=9
\end{aligned}
$$

Accepted Answer

A)  $ (7.9,2.4,9.0,12.3) $ 
B)  $ (3.6,1.1,4.1,5.6) $ 
C)  $ (3.2,1.0,3.7,5.0) $ 
D)  $ (4.3,1.3,4.9,6.7) $ 
A)  $ (7.9,2.4,9.0,12.3) $ 
B)  $ (3.6,1.1,4.1,5.6) $ 
C)  $ (3.2,1.0,3.7,5.0) $ 
D)  $ (4.3,1.3,4.9,6.7) $

Question 18

Use the Gauss-Jordan method to solve the system of equations.
-$
\begin{array}{l}
x-y+z=4 \
x+y+z=-2 \
x+y-z=-12
\end{array}
$

Accepted Answer

A)  $ (-4,5,-3) $ 
B)  $ (-4,-3,5) $ 
C)  $ (5,-4,-3) $ 
D)  No solution 
A)  $ (-4,5,-3) $ 
B)  $ (-4,-3,5) $ 
C)  $ (5,-4,-3) $ 
D)  No solution

Question 19

Solve the problem.
-Suppose the following matrix represents the input-output matrix, T, of a simplified economy.   Find the amount of each commodity that should be produced.

Accepted Answer

A) 3360 units of manufacturing, 2796 units of agriculture, and 3012 units of transportation. 
B) 2976 units of manufacturing, 3386.4 units of agriculture, and 2791.2 units of transportation. 
C) 3012 units of manufacturing, 3360 units of agriculture, and 2791.2 units of transportation. 
D) 3012 units of manufacturing, 2796 units of agriculture, and 2976 units of transportation. 
A) 3360 units of manufacturing, 2796 units of agriculture, and 3012 units of transportation. 
B) 2976 units of manufacturing, 3386.4 units of agriculture, and 2791.2 units of transportation. 
C) 3012 units of manufacturing, 3360 units of agriculture, and 2791.2 units of transportation. 
D) 3012 units of manufacturing, 2796 units of agriculture, and 2976 units of transportation.

Question 20

Solve the system of equations by using the inverse of the coefficient matrix.
-$$
\begin{array}{l}
9 x-2 y+5 z=78 \
2 x+3 y-9 z=-50 \
3 x-8 y+6 z=49
\end{array}
$$

Accepted Answer

A)  $ (6,-1,7) $ 
B)  $ (1,7,1) $ 
C)  $ (5,-1,-7) $ 
D)  $ (5,1,7) $ 
A)  $ (6,-1,7) $ 
B)  $ (1,7,1) $ 
C)  $ (5,-1,-7) $ 
D)  $ (5,1,7) $

Find the ratios of products A, B, and C using a closed model. -

Perform the indicated operation, where possible. - $\left[\begin{array}{rr}-9 & 1 \\ 2 & 5\end{array}\right]+\left[\begin{array}{ll}6 & 2 \\ 6 & 3\end{array}\right]$

Find the ratios of products A, B, and C using a closed model. -

Use the Gauss-Jordan method to solve the system of equations. -2x-8y=6 -4x-16y=2

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist. -A is $2 \times 1$ , and $B$ is $1 \times 1$ .

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist. - $A$ is $4 \times 2$ , and B is $4 \times 2$ .

Does a matrix with a column of all zeros have an inverse? Why or why not?

Use the indicated row operation to change the matrix. -Replace $\mathrm{R}_{2}$ by $\mathrm{R}_{1}+(-1) \mathrm{R}_{2}$ . $\left[\begin{array}{rr|r}1 & -3 & 4 \\ 2 & 3 & 1\end{array}\right]$

Solve the problem. -A company makes three chocolate candies: cherry, almond, and raisin. Matrix A gives the amount of ingredients in one batch. Matrix B gives the costs of ingredients from suppliers J and K. What is The cost of 100 batches of each candy using ingredients from supplier K?

Find the matrix product, if possible. - $\left[\begin{array}{rrr}1 & 3 & -1 \\3 & 0 & 5\end{array}\right]\left[\begin{array}{rr}3 & 0 \\-1 & 1 \\0 & 5\end{array}\right]$

Find the ratios of products A, B, and C using a closed model. -

Find the values of the variables in the equation. - $\left[\begin{array}{crr}t-1 & -4 & 0 \\-3 & 4 & 3\end{array}\right]=\left[\begin{array}{lll}-4 & -4 & 0 \\-3 & x-8 & 3\end{array}\right]$

Use the echelon method to solve the system. +=1 -=-

Perform the indicated operation, where possible. - $\left[\begin{array}{rr}-1 & 5 \\ 0 & 4 \\ 8 & -4\end{array}\right]-\left[\begin{array}{ll}2 & 1 \\ 7 & 4 \\ 2 & 2\end{array}\right]$

Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.) - 1.5x+3.1y-2.3z+0.6w =2.7 6.1x-0.5z-3.4w =0.8 11.5y-2.2w =0 4.0x+3.0y-2.1z =9

Use the Gauss-Jordan method to solve the system of equations. - x-y+z=4 x+y+z=-2 x+y-z=-12

Solve the problem. -Suppose the following matrix represents the input-output matrix, T, of a simplified economy. Find the amount of each commodity that should be produced.

Solve the system of equations by using the inverse of the coefficient matrix. - 9x-2y+5z=78 2x+3y-9z=-50 3x-8y+6z=49

Linear Functions

Filters

Exam 2: Systems of Linear Equations and Matrices

Find the ratios of products A, B, and C using a closed model. -

Perform the indicated operation, where possible. - [−9125]+[6263] \left[\begin{array}{rr}-9 & 1 \\ 2 & 5\end{array}\right]+\left[\begin{array}{ll}6 & 2 \\ 6 & 3\end{array}\right] [−92​15​]+[66​23​]

Find the ratios of products A, B, and C using a closed model. -

Use the Gauss-Jordan method to solve the system of equations. -2x-8y=6 -4x-16y=2

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist. -A is 2×1 2 \times 1 2×1 , and B B B is 1×1 1 \times 1 1×1 .

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist. - A A A is 4×2 4 \times 2 4×2 , and B is 4×2 4 \times 2 4×2 .

Does a matrix with a column of all zeros have an inverse? Why or why not?

Use the indicated row operation to change the matrix. -Replace R2 \mathrm{R}_{2} R2​ by R1+(−1)R2 \mathrm{R}_{1}+(-1) \mathrm{R}_{2} R1​+(−1)R2​ . [1−34231] \left[\begin{array}{rr|r}1 & -3 & 4 \\ 2 & 3 & 1\end{array}\right] [12​−33​41​]

Solve the problem. -A company makes three chocolate candies: cherry, almond, and raisin. Matrix A gives the amount of ingredients in one batch. Matrix B gives the costs of ingredients from suppliers J and K. What is The cost of 100 batches of each candy using ingredients from supplier K?

Find the matrix product, if possible. - [13−1305][30−1105]\left[\begin{array}{rrr}1 & 3 & -1 \\3 & 0 & 5\end{array}\right]\left[\begin{array}{rr}3 & 0 \\-1 & 1 \\0 & 5\end{array}\right][13​30​−15​]​3−10​015​​

Find the ratios of products A, B, and C using a closed model. -

Find the values of the variables in the equation. - [t−1−40−343]=[−4−40−3x−83]\left[\begin{array}{crr}t-1 & -4 & 0 \\-3 & 4 & 3\end{array}\right]=\left[\begin{array}{lll}-4 & -4 & 0 \\-3 & x-8 & 3\end{array}\right][t−1−3​−44​03​]=[−4−3​−4x−8​03​]

Use the echelon method to solve the system. +=1 -=-

Perform the indicated operation, where possible. - [−15048−4]−[217422] \left[\begin{array}{rr}-1 & 5 \\ 0 & 4 \\ 8 & -4\end{array}\right]-\left[\begin{array}{ll}2 & 1 \\ 7 & 4 \\ 2 & 2\end{array}\right] ​−108​54−4​​−​272​142​​

Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.) - 1.5x+3.1y-2.3z+0.6w =2.7 6.1x-0.5z-3.4w =0.8 11.5y-2.2w =0 4.0x+3.0y-2.1z =9

Use the Gauss-Jordan method to solve the system of equations. - x-y+z=4 x+y+z=-2 x+y-z=-12

Solve the problem. -Suppose the following matrix represents the input-output matrix, T, of a simplified economy. Find the amount of each commodity that should be produced.

Solve the system of equations by using the inverse of the coefficient matrix. - 9x-2y+5z=78 2x+3y-9z=-50 3x-8y+6z=49

Linear Functions

Filters

Perform the indicated operation, where possible. - $\left[\begin{array}{rr}-9 & 1 \\ 2 & 5\end{array}\right]+\left[\begin{array}{ll}6 & 2 \\ 6 & 3\end{array}\right]$

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist. -A is $2 \times 1$ , and $B$ is $1 \times 1$ .

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist. - $A$ is $4 \times 2$ , and B is $4 \times 2$ .

Use the indicated row operation to change the matrix. -Replace $\mathrm{R}_{2}$ by $\mathrm{R}_{1}+(-1) \mathrm{R}_{2}$ . $\left[\begin{array}{rr|r}1 & -3 & 4 \\ 2 & 3 & 1\end{array}\right]$

Find the matrix product, if possible. - $\left[\begin{array}{rrr}1 & 3 & -1 \\3 & 0 & 5\end{array}\right]\left[\begin{array}{rr}3 & 0 \\-1 & 1 \\0 & 5\end{array}\right]$

Find the values of the variables in the equation. - $\left[\begin{array}{crr}t-1 & -4 & 0 \\-3 & 4 & 3\end{array}\right]=\left[\begin{array}{lll}-4 & -4 & 0 \\-3 & x-8 & 3\end{array}\right]$

Perform the indicated operation, where possible. - $\left[\begin{array}{rr}-1 & 5 \\ 0 & 4 \\ 8 & -4\end{array}\right]-\left[\begin{array}{ll}2 & 1 \\ 7 & 4 \\ 2 & 2\end{array}\right]$