Question 1

Determine whether the two matrices are inverses of each other by computing their product.
-

Accepted Answer

A) True 
 B)False

Question 2

Determine whether the two matrices are inverses of each other by computing their product.
-$\left[\begin{array}{rr}10 & 1 \ -1 & 0\end{array}\right]$ and $\left[\begin{array}{rr}0 & 1 \ -1 & 10\end{array}\right]$

Accepted Answer

A) True 
 B)False

Question 3

Write the word or phrase that best completes each statement or answers the question
-Using the matrices $X=\left[\begin{array}{ll}x & y \ z & w\end{array}\right]$ and $L=\left[\begin{array}{cc}1 & m \ n & o\end{array}\right]$, verify that $X-L \neq L-X$ (matrix subtraction is not commutative).

Accepted Answer

The answer of Write the word or phrase that best...

Question 4

Write a system of equations and use the inverse of the coefficient matrix to solve the system.
-A basketball fieldhouse seats 15,000 . Courtside seats sell for $\$ 10$, endzone for $\$ 7$, and balcony for $\$ 4$. Total for a sell-out is $\$ 84,000$. If half the courtside and balcony and all endzone seats are sold, the total is $\$ 49,000$. How many of each type are there?

Accepted Answer

A)  3200 courtside
1800 endzone
10,000 balcony 
B)  3000 courtside
5000 endzone
7000 balcony 
C)  3000 courtside
2000 endzone
10,000 balcony 
D)  4000 courtside
3000 endzone
8000 balcony

Question 5

Solve the problem by writing and solving a suitable system of equations.
-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?

Accepted Answer

A)  Mike $16 \mathrm{hr}$, Joe $8 \mathrm{hr}$, Bill $8 \mathrm{hr}$ 
B)  Mike 8 hr, Joe 8 hr, Bill $16 \mathrm{hr}$ 
C)  Mike 8 hr, Joe 16 hr, Bill 8 hr 
D)  Mike 12 hr, Joe 10 hr, Bill $10 \mathrm{hr}$

Question 6

Perform the indicated operation.
-$\operatorname{Let} C=\left[\begin{array}{r}6 \ -2 \ 8\end{array}\right]$. Find $\left(\frac{1}{2}\right) C$.

Accepted Answer

A) $\left[\begin{array}{r}3 \ -2 \ 8\end{array}\right]$ 
B) $\left[\begin{array}{r}12 \ -4 \ 16\end{array}\right]$ 
C) $\left[\begin{array}{r}6 \ -1 \ 8\end{array}\right]$ 
D) $\left[\begin{array}{r}3 \ -1 \ 4\end{array}\right]$

Question 7

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.
-$-x-y-z=-6$
$x+y+z=0$
$\mathrm{x}+\mathrm{y}-\mathrm{z}=4$

Accepted Answer

A)  Dependent 
B)  Independent 
C)  Inconsistent

Question 8

Solve the system by using the inverse of the coefficient matrix.
-$x+3 y+4 z=18$
$4 y+3 z=22$
$\mathrm{z}=2$

Accepted Answer

A)  No solution 
B)  $(-2,4,2)$ 
C)  $(-2,2,4)$ 
D)  $(2,4,-2)$

Question 9

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 $3 \times 3$ matrix showing the required ingredients for each cake. Assign the cakes to the rows and the ingredients to the columns.

Accepted Answer

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) $\left[\begin{array}{lll}2 & 2 & 3 \ 4 & 1 & 1 \ 2 & 2 & 2\end{array}\right]$ 
D) $\left[\begin{array}{lll}2 & 2 & 2 \ 4 & 1 & 1 \ 2 & 2 & 3\end{array}\right]$

Question 10

Obtain an equivalent system by performing the stated elementary operation on the system.
-Replace the fourth equation by the sum of itself and 3 times the second equation
X-2 y+5 z-6 w=4 \
4 y-z+4 w=-5 \
3 y-4 z+2 w=-3 \
2 y-2 z-3 w=8

Accepted Answer

A) x-2 y+5 z-6 w & =4 \
4 y-z+4 w & =-5 \
3 y-4 z+2 w & =-3 \
14 y-5 z+9 w & =-7 
B) x-2 y+5 z-6 w= & 4 \
12 y-3 z+12 w= & -15 \
3 y-4 z+2 w= & -3 \
2 y-2 z-3 w= & 8 
C) x-2 y+5 z-6 w=4 \
4 y-z+4 w=-5 \
3 y-4 z+2 w=-3 \
12 y+3 z+9 w=-7 
D) x-2 y+5 z-6 w=4 \
4 y-z+4 w=-5 \
3 y-4 z+2 w=-3 \
-10 y+5 z-15 w=23

Question 11

Find the production matrix for the input-output and demand matrices.
-$\mathrm{A}=\left[\begin{array}{ll}0.2 & 0.1 \ 0.5 & 0.4\end{array}\right], \mathrm{D}=\left[\begin{array}{l}3 \ 4\end{array}\right]$

Accepted Answer

A) $\left[\begin{array}{r}6.4 \ 13.65\end{array}\right]$ 
B) $\left[\begin{array}{r}5.12 \ 10.92\end{array}\right]$ 
C) $\left[\begin{array}{r}4.67 \ 10.92\end{array}\right]$ 
D) $\left[\begin{array}{r}5.09 \ 5.875\end{array}\right]$

Question 12

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$.
-$2 x+y-z=2$
$x-3 y+2 z=1$
$7 \mathrm{x}-7 \mathrm{y}+4 \mathrm{z}=7$

Accepted Answer

A)  $(2,5,7)$ 
B)  $(1,0,0)$ 
C)  No solution 
D)  $\left(\frac{7+z}{7}, \frac{5}{7} z, z\right)$ for any real number $z$

Question 13

Given the matrices $A$ and $B$, find the matrix product $A B$.
-$A=\left[\begin{array}{rr}-1 & 3 \ 3 & 2\end{array}\right], B=\left[\begin{array}{ll}-2 & 0 \ -1 & 1\end{array}\right]$ Find $A B$.

Accepted Answer

A) $\left[\begin{array}{rr}2 & -6 \ -2 & -1\end{array}\right]$ 
B) $\left[\begin{array}{ll}3 & -1 \ 2 & -8\end{array}\right]$ 
C) $\left[\begin{array}{rr}2 & 0 \ -3 & 2\end{array}\right]$ 
D) $\left[\begin{array}{ll}-1 & 3 \ -8 & 2\end{array}\right]$

Question 14

Determine whether the given ordered set of numbers is a solution of the system of equations.
-$(3,-4,2.5)$
$4 \mathrm{x}-3 \mathrm{y}+\mathrm{z}=24.5$
$5 \mathrm{x}+2 \mathrm{z}=20$
$0.5 \mathrm{x}+2 \mathrm{y}-2 \mathrm{z}=-9.5$

Accepted Answer

A) True 
 B)False

Question 15

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system.
-$\frac{3 x}{8}-\frac{3 y}{5}=\frac{33}{80}$
$\frac{4 x}{7}+\frac{4 y}{5}=\frac{37}{35}$

Accepted Answer

A)  $\left(\frac{1}{4}, \frac{1}{2}\right)$ 
B)  $\left(\frac{3}{4}, \frac{1}{2}\right)$ 
C)  $\left(\frac{3}{2}, \frac{3}{4}\right)$ 
D)  $\left(\frac{3}{2}, \frac{1}{4}\right)$

Question 16

a system of linear equations containing more variables than equations is always dependent.

Accepted Answer

A) True 
 B)False

Question 17

Write an augmented matrix for the system of equations.
-$2 x+9 y+6 z=32$
$4 x+6 y+4 z=24$
$3 x+2 y+8 z=63$

Accepted Answer

A) 
$\left[\begin{array}{lll|l}
2 & 4 & 3 & 32 \
9 & 6 & 2 & 24 \
6 & 4 & 8 & 63
\end{array}\right]$ 
B) 
$\left[\begin{array}{lll|l}2 & 9 & 6 & 32 \ 4 & 6 & 4 & 24 \ 3 & 2 & 8 & 63\end{array}\right]$ 
C) 
$\left[\begin{array}{lll}2 & 9 & 6 \ 4 & 6 & 4 \ 3 & 2 & 8\end{array}\right]$ 
D) 
$\left[\begin{array}{lll|l}32 & 6 & 9 & 2 \ 24 & 4 & 6 & 4 \ 63 & 8 & 2 & 3\end{array}\right]$

Question 18

Solve the problem.
-Suppose the following matrix represents the input-output matrix of a primitive economy. How much of each commodity should be produced to produce 89 bushels of yams and 60 pigs?

Accepted Answer

A) $X=\left[\begin{array}{c}201 \ 89.75\end{array}\right]$ 
B) $X=\left[\begin{array}{c}148.5 \ 134.25\end{array}\right]$ 
C) $X=\left[\begin{array}{c}178.5 \ 89.75\end{array}\right]$ 
D) $X=\left[\begin{array}{c}148.5 \ 167.63\end{array}\right]$

Question 19

The diagram shows the roads connecting four cities. Write a matrix to represent the number of routes between each pair of cities without passing through another city.

Accepted Answer

A)   
B)   
C)   
D)

Question 20

The diagram shows the roads connecting four cities.
 How many ways are there to travel between cities $\mathrm{Y}$ and $\mathrm{Z}$ by passing through exactly two cities? (Hint: Write a matrix, A, to represent the number of routes between each pair of cities without passing through another city. Then calculate $\mathrm{A}^{2}$ and $\mathrm{A}^{3}$ ).

Accepted Answer

A)  11 
B)  15 
C)  14 
D)  13

Exam 6: Systems of Linear Equations and Matrices

Determine whether the two matrices are inverses of each other by computing their product. -

Determine whether the two matrices are inverses of each other by computing their product. - $\left[\begin{array}{rr}10 & 1 \\ -1 & 0\end{array}\right]$ and $\left[\begin{array}{rr}0 & 1 \\ -1 & 10\end{array}\right]$

Write the word or phrase that best completes each statement or answers the question -Using the matrices $X=\left[\begin{array}{ll}x & y \\ z & w\end{array}\right]$ and $L=\left[\begin{array}{cc}1 & m \\ n & o\end{array}\right]$ , verify that $X-L \neq L-X$ (matrix subtraction is not commutative).

Perform the indicated operation. - $\operatorname{Let} C=\left[\begin{array}{r}6 \\ -2 \\ 8\end{array}\right]$ . Find $\left(\frac{1}{2}\right) C$ .

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent. - $-x-y-z=-6$ $x+y+z=0$ $\mathrm{x}+\mathrm{y}-\mathrm{z}=4$

Solve the system by using the inverse of the coefficient matrix. - $x+3 y+4 z=18$ $4 y+3 z=22$ $\mathrm{z}=2$

Obtain an equivalent system by performing the stated elementary operation on the system. -Replace the fourth equation by the sum of itself and 3 times the second equation X-2 y+5 z-6 w=4 \\ 4 y-z+4 w=-5 \\ 3 y-4 z+2 w=-3 \\ 2 y-2 z-3 w=8

Find the production matrix for the input-output and demand matrices. - $\mathrm{A}=\left[\begin{array}{ll}0.2 & 0.1 \\ 0.5 & 0.4\end{array}\right], \mathrm{D}=\left[\begin{array}{l}3 \\ 4\end{array}\right]$

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ . - $2 x+y-z=2$ $x-3 y+2 z=1$ $7 \mathrm{x}-7 \mathrm{y}+4 \mathrm{z}=7$

Given the matrices $A$ and $B$ , find the matrix product $A B$ . - $A=\left[\begin{array}{rr}-1 & 3 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{ll}-2 & 0 \\ -1 & 1\end{array}\right]$ Find $A B$ .

Determine whether the given ordered set of numbers is a solution of the system of equations. - $(3,-4,2.5)$ $4 \mathrm{x}-3 \mathrm{y}+\mathrm{z}=24.5$ $5 \mathrm{x}+2 \mathrm{z}=20$ $0.5 \mathrm{x}+2 \mathrm{y}-2 \mathrm{z}=-9.5$

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system. - $\frac{3 x}{8}-\frac{3 y}{5}=\frac{33}{80}$ $\frac{4 x}{7}+\frac{4 y}{5}=\frac{37}{35}$

a system of linear equations containing more variables than equations is always dependent.

Write an augmented matrix for the system of equations. - $2 x+9 y+6 z=32$ $4 x+6 y+4 z=24$ $3 x+2 y+8 z=63$

The diagram shows the roads connecting four cities. Write a matrix to represent the number of routes between each pair of cities without passing through another city.

Exam 6: Systems of Linear Equations and Matrices

Determine whether the two matrices are inverses of each other by computing their product. -

Determine whether the two matrices are inverses of each other by computing their product. - [101−10]\left[\begin{array}{rr}10 & 1 \\ -1 & 0\end{array}\right][10−1​10​] and [01−110]\left[\begin{array}{rr}0 & 1 \\ -1 & 10\end{array}\right][0−1​110​]

Perform the indicated operation. - Let⁡C=[6−28]\operatorname{Let} C=\left[\begin{array}{r}6 \\ -2 \\ 8\end{array}\right]LetC=​6−28​​ . Find (12) C\left(\frac{1}{2}\right) C(21​) C .

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent. - −x−y−z=−6-x-y-z=-6−x−y−z=−6 x+y+z=0x+y+z=0x+y+z=0 x+y−z=4\mathrm{x}+\mathrm{y}-\mathrm{z}=4x+y−z=4

Solve the system by using the inverse of the coefficient matrix. - x+3y+4z=18x+3 y+4 z=18x+3y+4z=18 4y+3z=224 y+3 z=224y+3z=22 z=2\mathrm{z}=2z=2

Obtain an equivalent system by performing the stated elementary operation on the system. -Replace the fourth equation by the sum of itself and 3 times the second equation X-2 y+5 z-6 w=4 \\ 4 y-z+4 w=-5 \\ 3 y-4 z+2 w=-3 \\ 2 y-2 z-3 w=8

Find the production matrix for the input-output and demand matrices. - A=[0.20.10.50.4],D=[34]\mathrm{A}=\left[\begin{array}{ll}0.2 & 0.1 \\ 0.5 & 0.4\end{array}\right], \mathrm{D}=\left[\begin{array}{l}3 \\ 4\end{array}\right]A=[0.20.5​0.10.4​],D=[34​]

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter z\mathrm{z}z . - 2x+y−z=22 x+y-z=22x+y−z=2 x−3y+2z=1x-3 y+2 z=1x−3y+2z=1 7x−7y+4z=77 \mathrm{x}-7 \mathrm{y}+4 \mathrm{z}=77x−7y+4z=7

Given the matrices AAA and BBB , find the matrix product ABA BAB . - A=[−1332],B=[−20−11]A=\left[\begin{array}{rr}-1 & 3 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{ll}-2 & 0 \\ -1 & 1\end{array}\right]A=[−13​32​],B=[−2−1​01​] Find ABA BAB .

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system. - 3x8−3y5=3380\frac{3 x}{8}-\frac{3 y}{5}=\frac{33}{80}83x​−53y​=8033​ 4x7+4y5=3735\frac{4 x}{7}+\frac{4 y}{5}=\frac{37}{35}74x​+54y​=3537​

a system of linear equations containing more variables than equations is always dependent.

Write an augmented matrix for the system of equations. - 2x+9y+6z=322 x+9 y+6 z=322x+9y+6z=32 4x+6y+4z=244 x+6 y+4 z=244x+6y+4z=24 3x+2y+8z=633 x+2 y+8 z=633x+2y+8z=63

Solve the problem. -Suppose the following matrix represents the input-output matrix of a primitive economy. How much of each commodity should be produced to produce 89 bushels of yams and 60 pigs?

The diagram shows the roads connecting four cities. Write a matrix to represent the number of routes between each pair of cities without passing through another city.

Determine whether the two matrices are inverses of each other by computing their product. - $\left[\begin{array}{rr}10 & 1 \\ -1 & 0\end{array}\right]$ and $\left[\begin{array}{rr}0 & 1 \\ -1 & 10\end{array}\right]$

Perform the indicated operation. - $\operatorname{Let} C=\left[\begin{array}{r}6 \\ -2 \\ 8\end{array}\right]$ . Find $\left(\frac{1}{2}\right) C$ .

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent. - $-x-y-z=-6$ $x+y+z=0$ $\mathrm{x}+\mathrm{y}-\mathrm{z}=4$

Solve the system by using the inverse of the coefficient matrix. - $x+3 y+4 z=18$ $4 y+3 z=22$ $\mathrm{z}=2$

Find the production matrix for the input-output and demand matrices. - $\mathrm{A}=\left[\begin{array}{ll}0.2 & 0.1 \\ 0.5 & 0.4\end{array}\right], \mathrm{D}=\left[\begin{array}{l}3 \\ 4\end{array}\right]$

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ . - $2 x+y-z=2$ $x-3 y+2 z=1$ $7 \mathrm{x}-7 \mathrm{y}+4 \mathrm{z}=7$

Given the matrices $A$ and $B$ , find the matrix product $A B$ . - $A=\left[\begin{array}{rr}-1 & 3 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{ll}-2 & 0 \\ -1 & 1\end{array}\right]$ Find $A B$ .

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system. - $\frac{3 x}{8}-\frac{3 y}{5}=\frac{33}{80}$ $\frac{4 x}{7}+\frac{4 y}{5}=\frac{37}{35}$

Write an augmented matrix for the system of equations. - $2 x+9 y+6 z=32$ $4 x+6 y+4 z=24$ $3 x+2 y+8 z=63$