Question 1

Solve the system of equations by using the inverse of the coefficient matrix.
-$
\begin{array}{rr}
x-2 y+3 z & =3 \
y-z+w & =-6 \
-2 x+2 y-2 z+4 w & =-18 \
2 y-3 z+w & =-10
\end{array}
$
A) $ (-2,3,3,-6) $
B) $ (-2,2,3,-5) $
C) $ (-3,2,4,-5) $
D) $ (-1,1,2,-4) $

Accepted Answer

The answer of Solve the system of equations by using...

Question 2

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] $
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

Accepted Answer

To subtract two matrices, subtract corresponding elements. For the given matrices, the subtraction is performed as follows: $(-1-2, 5-1$, $0-7, 4-4$, $8-2, -4-2)$, resulting in $(-3, 4$, $-7, 0$, $6, -6)$.

Question 3

Use graphing calculator to find the inverse of the matrix. Give 5 decimal places.
-$$
A=\left[\begin{array}{lll}
1.52 & 5.55 & 4 \
-0.63 & 7.33 & 3.21 \
8.2 & 0.003 & -2.8
\end{array}\right]
$$
A) $ \left[\begin{array}{lll}0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0\end{array}\right] $
B) $ \left[\begin{array}{rlr}0.65789 & 0.18018 & 0.25 \ -1.58730 & 0.13643 & 0.31153 \ 0.12195 & 333.33333 & -0.35714\end{array}\right] $ 
C) $ \left[\begin{array}{rrr}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}{rrr}0.15171 & -0.11491 & 0.08500 \ -0.18145 & 0.27379 & 0.05467 \ 0.44411 & -0.33622 & -0.10815\end{array}\right] $

Accepted Answer

Using a graphing calculator to find the inverse of the given matrix $A$, the correct inverse matrix is the one provided in option D, with each element rounded to five decimal places. Calculating the inverse of a matrix involves complex operations that are best handled by computational tools for accuracy, especially with non-integer values.

Question 4

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]
$
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

Accepted Answer

The answer of Find the matrix product, if possible.
-\(
\left[\begin{array}{rrr}
1 &...

Question 5

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 R and S. What is The cost of 100 batches of each candy using ingredients from supplier S?

A) $ \$ 16,500 $
B) $ \$ 6600 $
C) $ \$ 13,200 $
D) $ \$ 12,000 $

Accepted Answer

The answer of Solve the problem.
-A company makes three chocolate...

Question 6

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 X and Y. What is The cost of 100 batches of each candy using ingredients from supplier Y?

A) $ \$ 6600 $
B) $ \$ 7800 $
C) $ \$ 3300 $
D) $ \$ 4800 $

Accepted Answer

The answer of Solve the problem.
-A company makes three chocolate...

Question 7

Provide an appropriate response.
-If A is a 5 × 2 matrix and A+ K = A, what can you say about K?
A)K is a 2 × 5 matrix and each of its entries is the negative of the corresponding entry of A.
B)K is a 2 × 5 matrix and all entries of K are equal to 0.
C)K is a 5 × 2 matrix and all entries of K are equal to 0.
D)K is a 5 × 2 matrix and each of its entries is the negative of the corresponding entry of A.

Accepted Answer

For the equation A + K = A to hold true, K must be a matrix where all entries are 0, and it must have the same dimensions as A, which are 5 × 2. This ensures that adding K to A leaves A unchanged.

Question 8

Use the Gauss-Jordan method to solve the system of equations.
-$
\begin{array}{l}
8 x+9 y-z=48 \
x-4 y+7 z=65 \
2 x+y+z=22
\end{array}
$
A) $ (6,1,9) $
B) $ (6,9,1) $
C) $ (-6,1,12) $
D) No solution

Accepted Answer

To solve the system of equations using the Gauss-Jordan method, we first write the augmented matrix of the system and then apply row operations to transform it into reduced row echelon form (RREF). The system of equations is:$$ \begin{align*}8x + 9y - z &= 48 \x - 4y + 7z &= 65 \2x + y + z &= 22\end{align*} $$The augmented matrix is:$$ \left[ \begin{array}{ccc|c}8 & 9 & -1 & 48 \1 & -4 & 7 & 65 \2 & 1 & 1 & 22\end{array} \right] $$Applying row operations to simplify this matrix into RREF gives us:$$ \left[ \begin{array}{ccc|c}1 & 0 & 0 & 6 \0 & 1 & 0 & 1 \0 & 0 & 1 & 9\end{array} \right] $$This corresponds to the solution $x = 6$, $y = 1$, and $z = 9$, which matches choice (A) $ (6,1,9) $.

Question 9

Solve the problem.
-Mike, Joe, and Bill are painting a fence. The painting can be finished if Mike and Joe work together for 4 hours and Bill works alone for 2 hours; or if Mike and Joe work together for 2 hours and Bill Works alone for 5 hours; or if Mike works alone for 6 hours, Joe works alone for 2 hours, and Bill Works alone for 1 hour. How much time does it take for each man working alone to complete the Painting?
A)Mike: 12 hr; Joe: 10 hr; Bill: 10 hr
B)Mike: 8 hr; Joe: 8 hr; Bill: 16 hr
C)Mike: 16 hr; Joe: 8 hr; Bill: 8 hr
D)Mike: 8 hr; Joe: 16 hr; Bill: 8 hr

Accepted Answer

Let the work done by Mike, Joe, and Bill in one hour be $m$, $j$, and $b$ respectively. From the given conditions, we have the following equations:1. $4(m + j) + 2b = 1$ (Mike and Joe for 4 hours plus Bill for 2 hours complete the work)2. $2(m + j) + 5b = 1$ (Mike and Joe for 2 hours plus Bill for 5 hours complete the work)3. $6m + 2j + b = 1$ (Mike for 6 hours, Joe for 2 hours, and Bill for 1 hour complete the work)Solving these equations simultaneously, we find that $m = \frac{1}{8}$, $j = \frac{1}{16}$, and $b = \frac{1}{8}$. Therefore, Mike takes 8 hours, Joe takes 16 hours, and Bill takes 8 hours to complete the painting working alone.

Question 10

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.

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] $

Accepted Answer

The answer of Write a matrix to display the information.
-Four...

Question 11

Solve the system of equations by using the inverse of the coefficient matrix if it exists and by the echelon method if the inverse doesn't exist.
-$$
\begin{array}{l}
3 x-6 y=-3 \
6 x-12 y=-9
\end{array}
$$
A) No inverse, no solution for system
B) $ (2,2) $
C) $ (-3,-9) $
D) No inverse, $ \left(-1-\frac{1}{2} y, y\right) $

Accepted Answer

The answer of Solve the system of equations by using...

Question 12

Solve the problem.
-An appliance dealer has two stores in the town of Washingwell. During a given week, they have a beginning inventory of 
Washing Dish
Machines Washers
$ \mathrm{B}=\left[\begin{array}{ll}44 & 25 \ 51 & 38\end{array}\right] \begin{array}{l}\text { Store 1 } \ \text { Store 2 }\end{array} $
a purchase matrix of
Washing Dish
Machines Washers
$ P=\left[\begin{array}{ll}5 & 3 \ 7 & 0\end{array}\right] \begin{array}{l}\text { Store 1 } \ \text { Store 2 }\end{array} $
and an ending inventory of
Washing Dish
Machines Washers
$ E=\left[\begin{array}{ll}8 & 1 \ 3 & 2\end{array}\right] \begin{array}{l}\text { Store 1 } \ \text { Store 2 }\end{array} $
Find the sales matrix.
A)
B) 
C) 
D)

Accepted Answer

The answer of Solve the problem.
-An appliance dealer has two...

Question 13

Perform the indicated operation, where possible.
-$
\left[\begin{array}{lll}
2 & 4
\end{array}\right]+\left[\begin{array}{c}
-4 \
9
\end{array}\right]
$
A) $ \left[\begin{array}{ll}-2 & 13\end{array}\right] $
B) $ \left[\begin{array}{c}-2 \ 13\end{array}\right] $
C) $ \left[\begin{array}{rr}2 & -4 \ 4 & 9\end{array}\right] $
D) Not possible

Accepted Answer

Matrix addition is only defined for matrices of the same dimensions. The given matrices have different dimensions, making the operation undefined.

Question 14

Solve the problem.
-A simplified economy has only two industries, the electric company and the gas company. Each dollar's worth of the electric company's output requires 0.20 of its own output and 0.4 of the gas Company's output. Each dollar's worth of the gas company's output requires 0.50 of its own output
And 0.7 of the electric company's output. Construct the input-output matrix. 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Solve the problem.
-A simplified economy has only...

Question 15

Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.)
-$ \left[\begin{array}{rrr}1 & 2 & 0 \ -2 & 0 & 2 \ 3 & 2 & -4\end{array}\right] $ and $ \left[\begin{array}{rrr}\frac{1}{2} & -1 & -\frac{1}{2} \ \frac{1}{4} & \frac{1}{2} & \frac{1}{4} \ \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2}\end{array}\right] $

Accepted Answer

Multiplying the two matrices results in the identity matrix $I_3 = \left[\begin{array}{rrr}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]$, confirming they are inverses of each other.

Question 16

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.)
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 $

Accepted Answer

Given the conditions, the total number of CDs is 60, with twice the number of rock CDs equal to three times the number of country CDs, and the number of jazz CDs equal to the number of classical CDs. By setting up equations based on these conditions and solving them with $w$ as the parameter, option C correctly represents the distribution of CDs among the four types, adhering to the given relationships and constraints.

Question 17

Use graphing calculator to find the inverse of the matrix. Give 5 decimal places.
-$$
A=\left[\begin{array}{ccc}
1 / 3 & 2 / 7 & 4 / 5 \
17 / 11 & 5 & 0 \
18 / 13 & -5 / 6 & 8 / 7
\end{array}\right]
$$
A) $ \left[\begin{array}{lll}0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0\end{array}\right] $
B) $ \left[\begin{array}{rrr}2.65952 & 0.15830 & -1.86167 \ -9.04238 & -0.33824 & 6.32966 \ 3.37129 & 0.05484 & -1.48490\end{array}\right] $
C) $ \left[\begin{array}{rrr}-1.10557 & 0.19216 & 0.77390 \ 0.34172 & 0.14061 & -0.23920 \ 1.58861 & -0.13028 & -0.23703\end{array}\right] $
D)
$ \left[\begin{array}{lll}3 & 3.5 & 1.25 \ 0.64706 & 5 & 0 \ 0.72222 & -1.2 & 1.14286\end{array}\right] $

Accepted Answer

The inverse of matrix A is calculated using a graphing calculator or matrix algebra, resulting in the matrix provided in option C.

Question 18

Solve the problem.
-Carole's car averages 15.3 miles per gallon in city driving and 24.5 miles per gallon in highway driving. If she drove a total of 382.7 miles on 19 gallons of gas, then how many of the gallons were
Used for city driving?
A)10 gallons
B)11 gallons
C)15 gallons
D)9 gallons

Accepted Answer

The answer of Solve the problem.
-Carole's car averages 15.3 miles...

Question 19

Find the value.
-Let $ \mathrm{A}=\left[\begin{array}{ll}-5 & 2\end{array}\right] $ and $ \mathrm{B}=\left[\begin{array}{ll}1 & 0\end{array}\right] ; 2 \mathrm{~A}+3 \mathrm{~B} $
A) $ \left[\begin{array}{ll}-2 & 2\end{array}\right] $
B) $ \left[\begin{array}{ll}-9 & 4\end{array}\right] $
C) $ [-74] $
D) $ [-104] $

Accepted Answer

The answer of Find the value.
-Let \( \mathrm{A}=\left[\begin{array}{ll}-5 & 2\end{array}\right]...

Question 20

-ABC is a 2 × 4 matrix.

Accepted Answer

The answer of 
-ABC is a 2 × 4 matrix....

Quiz 2: Systems of Linear Equations and Matrices

Solve the system of equations by using the inverse of the coefficient matrix. - $\begin{array}{rr}x-2 y+3 z & =3 \\y-z+w & =-6 \\-2 x+2 y-2 z+4 w & =-18 \\2 y-3 z+w & =-10\end{array}$

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

Use graphing calculator to find the inverse of the matrix. Give 5 decimal places. - $A=\left[\begin{array}{lll}1.52 & 5.55 & 4 \\-0.63 & 7.33 & 3.21 \\8.2 & 0.003 & -2.8\end{array}\right]$

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

Provide an appropriate response. -If A is a 5 × 2 matrix and A+ K = A, what can you say about K?

Use the Gauss-Jordan method to solve the system of equations. - $\begin{array}{l}8 x+9 y-z=48 \\x-4 y+7 z=65 \\2 x+y+z=22\end{array}$

Solve the system of equations by using the inverse of the coefficient matrix if it exists and by the echelon method if the inverse doesn't exist. - $\begin{array}{l}3 x-6 y=-3 \\6 x-12 y=-9\end{array}$

Perform the indicated operation, where possible. - $\left[\begin{array}{lll}2 & 4\end{array}\right]+\left[\begin{array}{c}-4 \\9\end{array}\right]$

Use graphing calculator to find the inverse of the matrix. Give 5 decimal places. - $A=\left[\begin{array}{ccc}1 / 3 & 2 / 7 & 4 / 5 \\17 / 11 & 5 & 0 \\18 / 13 & -5 / 6 & 8 / 7\end{array}\right]$

Solve the problem. -Carole's car averages 15.3 miles per gallon in city driving and 24.5 miles per gallon in highway driving. If she drove a total of 382.7 miles on 19 gallons of gas, then how many of the gallons were Used for city driving?

Find the value. -Let $\mathrm{A}=\left[\begin{array}{ll}-5 & 2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}1 & 0\end{array}\right] ; 2 \mathrm{~A}+3 \mathrm{~B}$

-ABC is a 2 × 4 matrix.

Quiz 2: Systems of Linear Equations and Matrices

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​​

Use graphing calculator to find the inverse of the matrix. Give 5 decimal places. - A=[1.525.554−0.637.333.218.20.003−2.8]A=\left[\begin{array}{lll}1.52 & 5.55 & 4 \\-0.63 & 7.33 & 3.21 \\8.2 & 0.003 & -2.8\end{array}\right]A=​1.52−0.638.2​5.557.330.003​43.21−2.8​​

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​​

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 R and S. What is The cost of 100 batches of each candy using ingredients from supplier S?

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 X and Y. What is The cost of 100 batches of each candy using ingredients from supplier Y?

Provide an appropriate response. -If A is a 5 × 2 matrix and A+ K = A, what can you say about K?

Use the Gauss-Jordan method to solve the system of equations. - 8x+9y−z=48x−4y+7z=652x+y+z=22\begin{array}{l}8 x+9 y-z=48 \\x-4 y+7 z=65 \\2 x+y+z=22\end{array}8x+9y−z=48x−4y+7z=652x+y+z=22​

Solve the system of equations by using the inverse of the coefficient matrix if it exists and by the echelon method if the inverse doesn't exist. - 3x−6y=−36x−12y=−9\begin{array}{l}3 x-6 y=-3 \\6 x-12 y=-9\end{array}3x−6y=−36x−12y=−9​

Perform the indicated operation, where possible. - [24]+[−49]\left[\begin{array}{lll}2 & 4\end{array}\right]+\left[\begin{array}{c}-4 \\9\end{array}\right][2​4​]+[−49​]

Use graphing calculator to find the inverse of the matrix. Give 5 decimal places. - A=[1/32/74/517/115018/13−5/68/7]A=\left[\begin{array}{ccc}1 / 3 & 2 / 7 & 4 / 5 \\17 / 11 & 5 & 0 \\18 / 13 & -5 / 6 & 8 / 7\end{array}\right]A=​1/317/1118/13​2/75−5/6​4/508/7​​

Solve the problem. -Carole's car averages 15.3 miles per gallon in city driving and 24.5 miles per gallon in highway driving. If she drove a total of 382.7 miles on 19 gallons of gas, then how many of the gallons were Used for city driving?

Find the value. -Let A=[−52] \mathrm{A}=\left[\begin{array}{ll}-5 & 2\end{array}\right] A=[−5​2​] and B=[10];2 A+3 B \mathrm{B}=\left[\begin{array}{ll}1 & 0\end{array}\right] ; 2 \mathrm{~A}+3 \mathrm{~B} B=[1​0​];2 A+3 B

-ABC is a 2 × 4 matrix.

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

Use graphing calculator to find the inverse of the matrix. Give 5 decimal places. - $A=\left[\begin{array}{lll}1.52 & 5.55 & 4 \\-0.63 & 7.33 & 3.21 \\8.2 & 0.003 & -2.8\end{array}\right]$

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

Use the Gauss-Jordan method to solve the system of equations. - $\begin{array}{l}8 x+9 y-z=48 \\x-4 y+7 z=65 \\2 x+y+z=22\end{array}$

Solve the system of equations by using the inverse of the coefficient matrix if it exists and by the echelon method if the inverse doesn't exist. - $\begin{array}{l}3 x-6 y=-3 \\6 x-12 y=-9\end{array}$

Perform the indicated operation, where possible. - $\left[\begin{array}{lll}2 & 4\end{array}\right]+\left[\begin{array}{c}-4 \\9\end{array}\right]$

Use graphing calculator to find the inverse of the matrix. Give 5 decimal places. - $A=\left[\begin{array}{ccc}1 / 3 & 2 / 7 & 4 / 5 \\17 / 11 & 5 & 0 \\18 / 13 & -5 / 6 & 8 / 7\end{array}\right]$

Find the value. -Let $\mathrm{A}=\left[\begin{array}{ll}-5 & 2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}1 & 0\end{array}\right] ; 2 \mathrm{~A}+3 \mathrm{~B}$