Question 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 $.
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 $

Accepted Answer

For matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix. Since both A and B are $4 \times 2$, neither AB nor BA can be formed because the number of columns in A does not match the number of rows in B, and vice versa.

Question 2

Use a graphing calculator to find the matrix product and/or sum.
-Find $ A B+A C $.
$$
A=\left[\begin{array}{rrrr}
17 & -14 & 18 & 18 \
30 & 5 & -7 & 2 \
11 & 3 & -2 & 1 \
-5 & 32 & 5 & 10
\end{array}\right] B=\left[\begin{array}{rrr}
5 & 17 & 12 \
7 & 6 & -4 \
-12 & -9 & 15 \
3 & -2 & -14
\end{array}\right] C=\left[\begin{array}{rrr}
7 & -4 & 3 \
-16 & -19 & 20 \
-6 & -15 & -8 \
5 & 16 & 17
\end{array}\right]
$$
A) $ \left[\begin{array}{rrr}150 & 3 & 281 \ 457 & 521 & 487 \ 97 & 206 & 88 \ 174 & -461 & 502\end{array}\right] $
B) $ \left[\begin{array}{rrr}-168 & 3 & 281 \ 259 & 580 & 297 \ 97 & 206 & 88 \ 174 & 58 & -28\end{array}\right] $
C)
$ \left[\begin{array}{rrr}150 & 223 & 211 \ 457 & 521 & 487 \ 149 & 166 & 202 \ -358 & -461 & 502\end{array}\right] $
D) Does not exist

Accepted Answer

The answer of Use a graphing calculator to find the...

Question 3

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 $.
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.

Accepted Answer

For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Since A is $2 \times 1$ and B is $1 \times 1$, AB exists and will be $2 \times 1$. However, for BA, the number of columns in B (1) does not match the number of rows in A (2), so BA does not exist.

Question 4

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.
-$ -2 x-6 y=-2 $
 $ 2 x-y=-5 $
A) $ (-1,2) $
B) $ (1,-2) $
C) $ (-2,1) $
D) $ (2,-1) $

Accepted Answer

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

Question 5

Solve the problem.
-An investor has $400,000 to invest in stocks, bonds, and commodities. If he plans to put three times as much into stocks as in bonds, how can he distribute his money among the three types of Investments? (Let x denote the amount put into stocks, y the amount put into bonds, and z the Amount put into commodities. Let all amounts be in dollars, and let z be the parameter.) 
A) $ x=300,000-z / 2, y=100,000-z / 2,0 \leq z \leq 400,000 $
B) $ x=100,000-z / 2, y=300,000-z / 2,0 \leq z \leq 400,000 $
C) $ x=300,000-3 z / 4, y=100,000-z / 4,0 \leq z \leq 400,000 $
D) $ x=100,000-3 z / 4, y=300,000-z / 4,0 \leq z \leq 400,000 $

Accepted Answer

The answer of Solve the problem.
-An investor has $400,000 to...

Question 6

Solve the problem.
-Carney and Dobler sell auto and hazard insurance. Their sales, in dollars, for the months of July and August are given in the following matrices. Auto Hazard
July: $ \left[\begin{array}{ll}22,000 & 45,000 \ 19,000 & 27,000\end{array}\right] $ Carney
August: $ \left[\begin{array}{ll}25,000 & 44,000 \ 14,000 & 21,000\end{array}\right] $ Carney
Find a matrix that gives the increase (decrease) in sales by each salesman for each type of insurance from July to August.
A) $ \left[\begin{array}{rr}-3000 & 1000 \ 5000 & 6000\end{array}\right] $
B) $ [-80007000] $
C) $ \left[\begin{array}{rr}3000 & -1000 \ -5000 & -6000\end{array}\right] $
D) $ [8000-7000] $

Accepted Answer

To find the increase or decrease in sales for each type of insurance by each salesman from July to August, subtract the July sales matrix from the August sales matrix. This results in: $ \left[\begin{array}{ll}25,000 - 22,000 & 44,000 - 45,000 \ 14,000 - 19,000 & 21,000 - 27,000\end{array}\right] = \left[\begin{array}{rr}3000 & -1000 \ -5000 & -6000\end{array}\right] $.

Question 7

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 $ 3 \times 2 $, and B is $ 2 \times 3 $.
A) $ 2 \times 2 ; 3 \times 3 $
B) $ 3 \times 3 $; BA does not exist.
C) $ \mathrm{AB} $ does not exist: $ 2 \times 2 $
D) $ 3 \times 3 ; 2 \times 2 $

Accepted Answer

The product AB is $ 3 \times 3 $ because the inner dimensions (2) match, and the outer dimensions (3 and 3) determine the size. The product BA is $ 2 \times 2 $ because the inner dimensions (3) match, and the outer dimensions (2 and 2) determine the size.

Question 8

Find the inverse, if it exists, for the matrix.
-$$
\left[\begin{array}{rrr}
2 & -1 & 0 \
3 & -2 & 0 \
-2 & 3 & 1
\end{array}\right]
$$
A)$ \left[\begin{array}{rrr}2 & -1 & 0 \ 3 & 2 & 0 \ -5 & 4 & 1\end{array}\right] $
B)$ \left[\begin{array}{rrr}1 & -1 & 0 \ 3 & -2 & 1 \ -5 & 4 & -1\end{array}\right] $
C) $ \left[\begin{array}{rrr}2 & -1 & 0 \ 3 & -2 & 0 \ -5 & 4 & -1\end{array}\right] $
D)$ \left[\begin{array}{rrr}2 & -1 & 0 \ 3 & -2 & 0 \ -5 & 4 & 1\end{array}\right] $

Accepted Answer

To find the inverse of a matrix, one common method is to apply the Gauss-Jordan elimination on the augmented matrix [A|I], where A is the given matrix and I is the identity matrix. However, for this specific question, we can observe that the third column of the given matrix is already in the form of an identity matrix's third column. This suggests that the transformation affecting the first two columns does not impact the third column, implying that the inverse matrix will maintain the identity matrix's third column. The correct inverse matrix must therefore have a third column of [0, 0, 1]. Among the options, only option D has this property and also correctly adjusts the first two columns to reflect the inverse transformations needed to revert the effects of the original matrix on any vector.

Question 9

Find the matrix product, if possible.
-$ \left[\begin{array}{rr}3 & -1 \ 3 & 0\end{array}\right]\left[\begin{array}{rr}0 & -1 \ 3 & 6\end{array}\right] $
A) $ \left[\begin{array}{ll}0 & 1 \ 9 & 0\end{array}\right] $
B) $ \left[\begin{array}{rr}-3 & -9 \ 0 & -3\end{array}\right] $
C) $ \left[\begin{array}{rr}-3 & 0 \ 27 & -3\end{array}\right] $
D) $ \left[\begin{array}{rr}-9 & -3 \ -3 & 0\end{array}\right] $

Accepted Answer

To find the product of two matrices, multiply each row of the first matrix by each column of the second matrix and sum the products. For the given matrices:- The element in the first row, first column is $3*0 + (-1)*3 = -3$.- The element in the first row, second column is $3*(-1) + (-1)*6 = -3 - 6 = -9$.- The element in the second row, first column is $3*0 + 0*3 = 0$.- The element in the second row, second column is $3*(-1) + 0*6 = -3$.Thus, the product is $ \left[\begin{array}{rr}-3 & -9 \ 0 & -3\end{array}\right] $.

Question 10

Solve the problem.
-Anne and Nancy use a metal alloy that is 25.75% copper to make jewelry. How many ounces of a 19% alloy must be mixed with a 28% alloy to form 92 ounces of the desired alloy?
A)25 ounces
B)23 ounces
C)74 ounces
D)69 ounces

Accepted Answer

The answer of Solve the problem.
-Anne and Nancy use a...

Question 11

Find the inverse, if it exists, for the matrix.
-$$
A=\left[\begin{array}{rrrr}
2 & -3 & 0 & 2 \
6 & -6 & 0 & 4 \
6 & -12 & 0 & 6 \
0 & 0 & -2 & 0
\end{array}\right]
$$
A)
$$
\left[\begin{array}{cccc}
\frac{1}{2} & -\frac{1}{3} & 0 & \frac{1}{2} \
\frac{1}{6} & -\frac{1}{6} & 0 & \frac{1}{4} \
\frac{1}{6} & -\frac{1}{12} & 0 & \frac{1}{6} \
0 & 0 & -\frac{1}{2} & 0
\end{array}\right]

B) 
\left[\begin{array}{cccc}
-1 & 1 & 0 & 3 \
\frac{1}{2} & 0 & 0 & -\frac{1}{2} \
0 & -\frac{1}{3} & 0 & -\frac{1}{2} \
0 & 0 & -\frac{1}{2} & 0
\end{array}\right]
$$
C)$$
\left[\begin{array}{cccc}
-1 & \frac{1}{2} & 0 & 0 \
1 & 0 & -\frac{1}{3} & 0 \
0 & 0 & 0 & -\frac{1}{2} \
3-\frac{1}{2} & -\frac{1}{2} & 0
\end{array}\right]
$$
D) No inverse

Accepted Answer

The answer of Find the inverse, if it exists, for...

Question 12

Write a matrix to display the information.
-A bakery sells three types of cakes. Cake I requires 2 cups of flour, 2 cups of sugar, and 2 eggs. Cake II requires 4 cups of flour, 1 cup of sugar, and 1 egg. Cake III requires 2 cups of flour, 2 cups of Sugar, and 3 eggs. Make a 3 × 3 matrix showing the required ingredients for each cake. Assign the
Cakes to the rows and the ingredients to the columns. 
A) $ \left[\begin{array}{lll}2 & 1 & 3 \ 2 & 1 & 2 \ 2 & 4 & 2\end{array}\right] $
B) $ \left[\begin{array}{lll}2 & 4 & 2 \ 2 & 1 & 2 \ 2 & 1 & 3\end{array}\right] $
C) $ \left[\begin{array}{lll}2 & 2 & 2 \ 4 & 1 & 1 \ 2 & 2 & 3\end{array}\right] $
D) $ \left[\begin{array}{lll}2 & 2 & 3 \ 1 & 1 & 4 \ 2 & 2 & 2\end{array}\right] $

Accepted Answer

The matrix correctly assigns cakes to rows and ingredients (flour, sugar, eggs) to columns, matching the given requirements for each cake.

Question 13

Find the ratios of products A, B, and C using a closed model.
-
A) 32: 40: 13
B) $ 8: 10: 3 $
C) $ 8: 10: 40 $
D) $ 8: 13: 32 $

Accepted Answer

The answer of Find the ratios of products A, B,...

Question 14

Find the matrix product, if possible.
-$ \left[\begin{array}{rrr}3 & -2 & 1 \ 0 & 4 & -3\end{array}\right]\left[\begin{array}{rr}3 & 0 \ -2 & 1\end{array}\right] $
A) $ \left[\begin{array}{rr}9 & -6 \ -6 & 8 \ 3 & -5\end{array}\right] $
B) $ \left[\begin{array}{rrr}9 & -6 & 3 \ -6 & 8 & -5\end{array}\right] $
C) $ \left[\begin{array}{ll}9 & 0 \ 0 & 4\end{array}\right] $
D) Does not exist

Accepted Answer

The matrix product does not exist because the number of columns in the first matrix (3) does not match the number of rows in the second matrix (2).

Question 15

Provide an appropriate response.
-What are the elements of the third row of the following matrix? 42) $$
\left[\begin{array}{rrrr}
-2 & 1 & 3 & 4 \
-3 & -1 & 4 & 7 \
1 & -5 & 2 & 6
\end{array}\right]
$$
A) $ 1,-5,2,6 $
B) $ -2,1,3,4 $
C) $ 3,4,2 $
D) $ -2,-1,2 $

Accepted Answer

The elements of the third row of the given matrix are $ 1,-5,2,6 $, which matches option A.

Question 16

Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.)
-$ \left[\begin{array}{ll}-5 & 1 \ -7 & 1\end{array}\right] $ and $ \left[\begin{array}{cc}\frac{1}{2} & -\frac{1}{2} \ \frac{7}{2} & -\frac{5}{2}\end{array}\right] $

Accepted Answer

Multiplying the two matrices together yields the identity matrix $I = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right]$, confirming they are inverses of each other.

Question 17

Solve the system of equations by using the inverse of the coefficient matrix.
-$$
\begin{array}{l}
x-y+3 z=5 \
2 x+z=1 \
x+4 y+z=-7
\end{array}
$$
A) $ (1,-2,0) $
B) $ (1,0,-2) $
C) $ (0,-2,1) $
D) No inverse, no solution for system

Accepted Answer

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

Question 18

Find the ratios of products A, B, and C using a closed model.
-
A) $ 33: 24: 10 $
B) $ 55: 76: 23 $
C) $ 2: 3: 3 $
D) $ 8: 11: 10 $

Accepted Answer

The answer of Find the ratios of products A, B,...

Question 19

Find the matrix product, if possible.
-$$
\left[\begin{array}{rr}
-1 & 3 \
5 & 4
\end{array}\right]\left[\begin{array}{lll}
0 & -2 & 4 \
1 & -3 & 2
\end{array}\right]
$$
A) $ \left[\begin{array}{rrr}3 & -7 & 2 \ 4 & -22 & 28\end{array}\right] $
B) $ \left[\begin{array}{rrr}3 & 4 & -7 \ -22 & 2 & 28\end{array}\right] $
C) $ \left[\begin{array}{rr}0 & -6 \ 12 & 5 \ -12 & 8\end{array}\right] $
D) Does not exist

Accepted Answer

To find the matrix product, multiply each row of the first matrix by each column of the second matrix and sum the products. For the given matrices, the calculation results in $ \left[\begin{array}{rrr}3 & -7 & 2 \ 4 & -22 & 28\end{array}\right] $.

Question 20

Find the inverse, if it exists, for the matrix.
-$$
\left[\begin{array}{rr}
8 & 2 \
-7 & 1
\end{array}\right]
$$
A) $ \left[\begin{array}{rr}1 & -2 \ 7 & 8\end{array}\right] $
B) 
C) 
D) No inverse

Accepted Answer

The answer of Find the inverse, if it exists, for...

Quiz 2: Systems of Linear Equations and Matrices

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 $4 \times 2$ .

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 $1 \times 1$ .

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. - $-2 x-6 y=-2$ $2 x-y=-5$

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 $3 \times 2$ , and B is $2 \times 3$ .

Find the inverse, if it exists, for the matrix. - $\left[\begin{array}{rrr}2 & -1 & 0 \\3 & -2 & 0 \\-2 & 3 & 1\end{array}\right]$

Find the matrix product, if possible. - $\left[\begin{array}{rr}3 & -1 \\ 3 & 0\end{array}\right]\left[\begin{array}{rr}0 & -1 \\ 3 & 6\end{array}\right]$

Solve the problem. -Anne and Nancy use a metal alloy that is 25.75% copper to make jewelry. How many ounces of a 19% alloy must be mixed with a 28% alloy to form 92 ounces of the desired alloy?

Find the inverse, if it exists, for the matrix. - $A=\left[\begin{array}{rrrr}2 & -3 & 0 & 2 \\6 & -6 & 0 & 4 \\6 & -12 & 0 & 6 \\0 & 0 & -2 & 0\end{array}\right]$

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

Find the matrix product, if possible. - $\left[\begin{array}{rrr}3 & -2 & 1 \\ 0 & 4 & -3\end{array}\right]\left[\begin{array}{rr}3 & 0 \\ -2 & 1\end{array}\right]$

Provide an appropriate response. -What are the elements of the third row of the following matrix? 42) $\left[\begin{array}{rrrr}-2 & 1 & 3 & 4 \\-3 & -1 & 4 & 7 \\1 & -5 & 2 & 6\end{array}\right]$

Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.) - $\left[\begin{array}{ll}-5 & 1 \\ -7 & 1\end{array}\right]$ and $\left[\begin{array}{cc}\frac{1}{2} & -\frac{1}{2} \\ \frac{7}{2} & -\frac{5}{2}\end{array}\right]$

Solve the system of equations by using the inverse of the coefficient matrix. - $\begin{array}{l}x-y+3 z=5 \\2 x+z=1 \\x+4 y+z=-7\end{array}$

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

Find the matrix product, if possible. - $\left[\begin{array}{rr}-1 & 3 \\5 & 4\end{array}\right]\left[\begin{array}{lll}0 & -2 & 4 \\1 & -3 & 2\end{array}\right]$

Find the inverse, if it exists, for the matrix. - $\left[\begin{array}{rr}8 & 2 \\-7 & 1\end{array}\right]$

Quiz 2: Systems of Linear Equations and Matrices

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 .

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 .

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. - −2x−6y=−2 -2 x-6 y=-2 −2x−6y=−2 2x−y=−5 2 x-y=-5 2x−y=−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 A A is 3×2 3 \times 2 3×2 , and B is 2×3 2 \times 3 2×3 .

Find the inverse, if it exists, for the matrix. - [2−103−20−231]\left[\begin{array}{rrr}2 & -1 & 0 \\3 & -2 & 0 \\-2 & 3 & 1\end{array}\right]​23−2​−1−23​001​​

Find the matrix product, if possible. - [3−130][0−136] \left[\begin{array}{rr}3 & -1 \\ 3 & 0\end{array}\right]\left[\begin{array}{rr}0 & -1 \\ 3 & 6\end{array}\right] [33​−10​][03​−16​]

Solve the problem. -Anne and Nancy use a metal alloy that is 25.75% copper to make jewelry. How many ounces of a 19% alloy must be mixed with a 28% alloy to form 92 ounces of the desired alloy?

Find the inverse, if it exists, for the matrix. - A=[2−3026−6046−120600−20]A=\left[\begin{array}{rrrr}2 & -3 & 0 & 2 \\6 & -6 & 0 & 4 \\6 & -12 & 0 & 6 \\0 & 0 & -2 & 0\end{array}\right]A=​2660​−3−6−120​000−2​2460​​

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

Find the matrix product, if possible. - [3−2104−3][30−21] \left[\begin{array}{rrr}3 & -2 & 1 \\ 0 & 4 & -3\end{array}\right]\left[\begin{array}{rr}3 & 0 \\ -2 & 1\end{array}\right] [30​−24​1−3​][3−2​01​]

Provide an appropriate response. -What are the elements of the third row of the following matrix? 42) [−2134−3−1471−526]\left[\begin{array}{rrrr}-2 & 1 & 3 & 4 \\-3 & -1 & 4 & 7 \\1 & -5 & 2 & 6\end{array}\right]​−2−31​1−1−5​342​476​​

Solve the system of equations by using the inverse of the coefficient matrix. - x−y+3z=52x+z=1x+4y+z=−7\begin{array}{l}x-y+3 z=5 \\2 x+z=1 \\x+4 y+z=-7\end{array}x−y+3z=52x+z=1x+4y+z=−7​

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

Find the matrix product, if possible. - [−1354][0−241−32]\left[\begin{array}{rr}-1 & 3 \\5 & 4\end{array}\right]\left[\begin{array}{lll}0 & -2 & 4 \\1 & -3 & 2\end{array}\right][−15​34​][01​−2−3​42​]

Find the inverse, if it exists, for the matrix. - [82−71]\left[\begin{array}{rr}8 & 2 \\-7 & 1\end{array}\right][8−7​21​]

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 $4 \times 2$ .

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 $1 \times 1$ .

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. - $-2 x-6 y=-2$ $2 x-y=-5$

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 $3 \times 2$ , and B is $2 \times 3$ .

Find the inverse, if it exists, for the matrix. - $\left[\begin{array}{rrr}2 & -1 & 0 \\3 & -2 & 0 \\-2 & 3 & 1\end{array}\right]$

Find the matrix product, if possible. - $\left[\begin{array}{rr}3 & -1 \\ 3 & 0\end{array}\right]\left[\begin{array}{rr}0 & -1 \\ 3 & 6\end{array}\right]$

Find the inverse, if it exists, for the matrix. - $A=\left[\begin{array}{rrrr}2 & -3 & 0 & 2 \\6 & -6 & 0 & 4 \\6 & -12 & 0 & 6 \\0 & 0 & -2 & 0\end{array}\right]$

Find the matrix product, if possible. - $\left[\begin{array}{rrr}3 & -2 & 1 \\ 0 & 4 & -3\end{array}\right]\left[\begin{array}{rr}3 & 0 \\ -2 & 1\end{array}\right]$

Provide an appropriate response. -What are the elements of the third row of the following matrix? 42) $\left[\begin{array}{rrrr}-2 & 1 & 3 & 4 \\-3 & -1 & 4 & 7 \\1 & -5 & 2 & 6\end{array}\right]$

Solve the system of equations by using the inverse of the coefficient matrix. - $\begin{array}{l}x-y+3 z=5 \\2 x+z=1 \\x+4 y+z=-7\end{array}$

Find the matrix product, if possible. - $\left[\begin{array}{rr}-1 & 3 \\5 & 4\end{array}\right]\left[\begin{array}{lll}0 & -2 & 4 \\1 & -3 & 2\end{array}\right]$

Find the inverse, if it exists, for the matrix. - $\left[\begin{array}{rr}8 & 2 \\-7 & 1\end{array}\right]$