Deck 6: Matrices and Determinants and Applications

Question 1

Choose the one alternative that best completes the statement or answers the question.
Write the augmented matrix for the given system.
\[\begin{array} { r }
x + 3 y + 5 z = - 3 \\
- 5 y + 3 z = - 8 \\
- 8 z = - 6

Answer 1

D

Question 2

Explain the meaning of the notation $3 \mathrm { R } _ { 1 } + \mathrm { R } _ { 2 } \rightarrow \mathrm { R } _ { 2 }$

Answer 2

Add 3 times row 1 to row 2 and replace the original row 2 with the result.

Question 3

Explain the meaning of the notation $4 \mathrm { R } _ { 2 } + \mathrm { R } _ { 3 } \rightarrow \mathrm { R } _ { 3 }$

Answer 3

Add 4 times row 2 to row 3 and replace the original row 3 with the result.

Question 4

A rectangular array of elements is called a .

Question 5

Choose the one alternative that best completes the statement or answers the question.
Write the augmented matrix for the given system.
$$\begin{aligned}- 3 x - 6 y + 9 z & = 12 \\- x + 2 y & = - 10 \\x - 8 z & = - 2\end{aligned}$$

A)
$$\left[ \begin{array} { r r r | r } - 3 & - 6 & 9 & 12 \\- 1 & 2 & 0 & - 10 \\1 & - 8 & 0 & - 2\end{array} \right]$$
B)
$$\left[ \begin{array} { r r r } - 3 & - 6 & 9 \\- 1 & 2 & 0 \\1 & 0 & - 8 \\12 & - 10 & - 2\end{array} \right]$$
C)
$$\left[ \begin{array} { r r r | r } - 3 & - 6 & 9 & 12 \\- 1 & 2 & 0 & - 10 \\1 & 0 & - 8 & - 2\end{array} \right]$$
D)
$$\left[ \begin{array} { r r r | r } - 3 & - 6 & 9 & 12 \\0 & - 1 & 2 & - 10 \\0 & 1 & - 8 & - 2\end{array} \right]$$

Question 6

Write a system of linear equations represented by the augmented matrix.
$$\left[ \begin{array} { r r r | r } 1 & 0 & 0 & 7 \\0 & 1 & 0 & - 9 \\0 & 0 & 1 & \frac { 1 } { 9 }\end{array} \right]$$

A) $x = - 7$
$y = 9$
$z = - \frac { 1 } { 9 }$
B) $x = - 7$
$y = - 9$
$z = \frac { 1 } { 9 }$
C) $x = 7$
$y = 9$
$z = \frac { 1 } { 9 }$
D) $x = 7$
$y = - 9$
$z = \frac { 1 } { 9 }$

Question 7

Choose the one alternative that best completes the statement or answers the question.
Write the augmented matrix for the given system.
$$\begin{aligned}7 x + 9 & = - 7 z \\- 4 x + 4 z & = - 8 - 6 y \\- 3 x - 7 y + 9 z & = 2\end{aligned}$$

A)
$\left[ \begin{array} { r r r | r } 7 & 0 & 7 & - 9 \\ - 4 & 6 & 4 & - 8 \\ - 3 & - 7 & 9 & 2 \end{array} \right]$
B) $\left[ \begin{array} { r r r | r } 7 & 0 & 7 & 0 \\ - 4 & - 6 & 4 & 0 \\ - 3 & - 7 & 9 & 0 \end{array} \right]$
C)
$\left[ \begin{array} { r r r | r } 7 & 0 & 9 & - 7 \\ - 4 & - 6 & 4 & - 8 \\ - 3 & - 7 & 9 & 2 \end{array} \right]$
D)
$$\left[ \begin{array} { r r r | r } 7 & 0 & - 7 & 9 \\- 4 & - 6 & 4 & - 8 \\- 3 & - 7 & 9 & 2\end{array} \right]$$

Question 8

Perform the elementary row operation on the given matrix.
$$3 R _ { 2 } + R _ { 1 } \rightarrow R _ { 1 }$$
$$\left[ \begin{array} { r r | r } 4 & 5 & 1 \\- 2 & - 7 & - 1\end{array} \right]$$

Question 9

Identify the elements on the main diagonal. $$\left[ \begin{array} { r r r | r } - 4 & - 1 & 1 & 8 \\2 & 0 & 5 & 11 \\0 & 1 & - 7 & - 6\end{array} \right]$$

Question 10

Explain the meaning of the notation $- R _ { 2 } \rightarrow R _ { 2 }$

Question 11

Explain the meaning of the notation $3 \mathrm { R } _ { 1 } \rightarrow \mathrm { R } _ { 1 }$

Question 12

An matrix is used to represent a system of linear equations written in standard form.

Question 13

Write a system of linear equations represented by the augmented matrix.
$$\left[ \begin{array} { r r r | r } 4 & 9 & - 4 & - 12 \\9 & 2 & 7 & 6 \\4 & - 2 & 2 & - 1\end{array} \right]$$

A) $4 x + 9 y - 4 z = 12$
$9 x + 2 y + 7 z = - 6$
$4 x - 2 y + 2 z = 1$
B) $4 x + 9 y + 4 z = - 12$
$9 x + 2 y - 2 z = 6$
$- 4 x + 7 y + 2 z = - 1$
C) $4 x + 9 y - 4 z = - 12$
$9 x + 2 y + 7 z = 6$
$4 x - 2 y + 2 z = - 1$
D) $4 x = - 12$
$2 y = 6$
$2 z = - 1$

Question 14

Explain the meaning of the notation $R _ { 2 } \Longleftrightarrow R _ { 3 }$

Question 15

Explain the meaning of $R _ { 1 } \Leftarrow R _ { 3 }$

Question 16

Perform the elementary row operation on the given matrix.
$$- \frac { 1 } { 2 } R _ { 2 } \rightarrow R _ { 2 }$$
$$\left[ \begin{array} { r r | r } 9 & 11 & 1 \\16 & 32 & 20\end{array} \right]$$

Question 17

Choose the one alternative that best completes the statement or answers the question.
Write the augmented matrix for the given system.
$$\begin{array} { l } 4 ( x - 3 y ) = 9 y + 6 \\8 x = 9 y + 6\end{array}$$

Question 18

Write a system of linear equations represented by the augmented matrix.
$$\left[ \begin{array} { r r | r } - 7 & 9 & - 1 \\5 & - 6 & 7\end{array} \right]$$

A) $- 7 x - 9 y = 1$
$5 x + 6 y = 7$
B) $- 7 x - 9 y = 1$
$5 x + 6 y = - 7$
C) $- 7 x + 9 y = - 1$
$5 x - 6 y = - 7$
D) $- 7 x + 9 y = - 1$
$5 x - 6 y = 7$

Question 19

Write a system of linear equations represented by the augmented matrix.
$$\left[ \begin{array} { r r | r } 5 & 8 & 6 \\0 & - 5 & 11\end{array} \right]$$

A)
$5 x + 8 y = 6$
[x - 5 y = 11\]

B)[5 x + 8 y = 6\]
(- 5 x = 11\)

C)(5 x = 6\)
(8 x - 5 y = 11\)

D)(5 x + 8 y = 6\)
(- 5 y = 11\)

Question 20

row operations performed on an augmented matrix results in a new augmented
matrix that represents an equivalent system of equations.

Question 21

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } 2 ( x - 2 z ) = 3 y + x + 30 \\x = 2 y - 2 z - 3 \\- 5 x + 2 y + 6 z = - 41\end{array}$$

A) $\{ ( 3 , - 1,5 ) \}$
B) $\{ ( 1 , - 3 , - 5 ) \}$
C) $\{ ( - 3,1 , - 5 ) \}$
D) $\{ ( - 1,3,5 ) \}$

Question 22

Perform the elementary row operation on the given matrix.
$$\begin{array} { l } R _ { 2 } \Leftrightarrow R _ { 3 } \\{ \left[ \begin{array} { r r r | r } - 8 & 2 & - 9 & - 4 \\- 3 & 7 & - 6 & 6 \\5 & 0 & 3 & 6\end{array} \right] }\end{array}$$

Question 23

Perform the indicated row operations, then write the new matrix.
$$\left[ \begin{array} { r r r | r } 1 & 1 & 1 & - 1 \\- 2 & 3 & 5 & 3 \\3 & 2 & 4 & 1\end{array} \right] \begin{array} { c } 2 R 1 + R 2 \rightarrow R 2 , \\- 3 R 1 + R 3 \rightarrow R 3\end{array}$$

A)
$$\left[ \begin{array} { r r r | r } 1 & 1 & 1 & - 1 \\0 & 5 & 7 & 1 \\0 & - 1 & 1 & 4\end{array} \right]$$
B)
$$\left[ \begin{array} { r r r | r } 1 & 1 & 1 & - 1 \\0 & 5 & 7 & 3 \\0 & - 1 & 1 & 1\end{array} \right]$$
C) $$\left[ \begin{array} { r r r | r } 1 & 1 & 1 & - 1 \\- 2 & 3 & 5 & 3 \\0 & - 1 & 1 & 1\end{array} \right]$$
D)
$\left[ \begin{array} { l l l | r } 1 & 1 & 1 & - 1 \\ 0 & 5 & 7 & 1 \\ 3 & 2 & 4 & 1 \end{array} \right]$

Question 24

Determine if the matrix is in row-echelon form.
$$\left[ \begin{array} { r r r | r } 1 & - 6 & - 8 & - 7 \\0 & 1 & 0 & 8 \\0 & 0 & 2 & 3\end{array} \right]$$

A) No
B) Yes

Question 25

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
True or false? A system of linear equations in three variables may have exactly two solutions.

Question 26

Find the partial fraction decomposition for the given rational expression. Use the technique of Gaussian
elimination to find A, B, and C.
$$\frac { - 3 x ^ { 2 } + 7 x - 10 } { ( x + 2 ) ( x - 1 ) ^ { 2 } } = \frac { A } { x + 2 } + \frac { B } { x - 1 } - \frac { C } { ( x - 1 ) ^ { 2 } }$$

A) $\frac { - 4 } { x + 2 } + \frac { 1 } { x - 1 } - \frac { 2 } { ( x - 1 ) ^ { 2 } }$
B) $\frac { - 4 } { x + 2 } + \frac { 2 } { x - 1 } - \frac { 2 } { ( x - 1 ) ^ { 2 } }$
C) $\frac { 4 } { x + 2 } + \frac { 1 } { x - 1 } - \frac { 2 } { ( x - 1 ) ^ { 2 } }$
D) $\frac { - 4 } { x + 2 } + \frac { 1 } { x - 1 } + \frac { 2 } { ( x - 1 ) ^ { 2 } }$

Question 27

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } - 5 y = 24 - 3 x \\2 ( x - 4 y ) = 49 - y\end{array}$$

A) $\{ ( 7,9 ) \}$
В) $\{ ( - 7 , - 9 ) \}$
C) $\{ ( 9,7 ) \}$
D) $\{ ( - 9 , - 7 ) \}$

Question 28

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } 5 x + 9 z = - 16 + 6 y \\8 x = 7 y + 7 z - 141 \\8 z = - 2 y - 2 x + 42\end{array}$$

A) $\{ ( 5 , - 8,6 ) \}$
B) $\{ ( 5,8 , - 6 ) \}$
C) $\{ ( 8 , - 5,6 ) \}$
D) $\{ ( - 8,5,6 ) \}$

Question 29

Use a calculator to approximate the reduced row-echelon form of the augmented matrix representing the
given system. Give the solution set where x, y, and z are rounded to 2 decimal places.
$$\begin{aligned}0.52 x - 3.79 y - 4.67 z & = 9.15 \\0.03 x + 0.06 y + 0.13 z & = 0.53 \\0.974 x + 0.813 y + 0.419 z & = 0.189\end{aligned}$$

A) $\{ ( 6.18 , - 11.2,7.82 ) \}$
В) $\{ ( - 15.61 , - 1.39 , - 2.84 ) \}$
C) $\{ ( 7.91 , - 1.03,3.18 ) \}$
D) $\{ ( - 4.49 , - 6.63,2.49 ) \}$

Question 30

Perform the elementary row operation on the given matrix.
$$- 5 R _ { 1 } + R _ { 3 } \rightarrow R _ { 3 }$$
$\left[ \begin{array} { r r r | r } 1 & 16 & 5 & 3 \\ 7 & 18 & 5 & 8 \\ 5 & 12 & 9 & 13 \end{array} \right]$

A) $\left[ \begin{array} { r r r | r } 1 & 16 & 5 & 3 \\ 2 & - 62 & - 20 & - 7 \\ 5 & 12 & 9 & 13 \end{array} \right]$ C) $\left[ \begin{array} { r r r | r } 1 & 16 & 5 & 3 \\ 7 & 18 & 5 & 8 \\ 0 & - 68 & - 16 & 16 \end{array} \right]$
В) $\left[ \begin{array} { r r r | r } 1 & 16 & 5 & 3 \\ 7 & 18 & 5 & 8 \\ 0 & - 68 & - 16 & - 2 \end{array} \right]$ D) $\left[ \begin{array} { r r r | r } 1 & 16 & 5 & 3 \\ 7 & 18 & 5 & 8 \\ 6 & 28 & 14 & 16 \end{array} \right]$

Question 31

Determine if the matrix is in row-echelon form.
$$\left[ \begin{array} { r r r r | r } 1 & 0 & 0 & 0 & 2 \\0 & 1 & 0 & 0 & 4 \\0 & 0 & - 1 & 0 & 9 \\0 & 0 & 0 & 1 & - 6\end{array} \right]$$

A) No
B) Yes

Question 32

Solve the problem.
Danielle stayed in three different cities (Washington, D.C., Atlanta, Georgia, and Dallas, Texas) for a total of 22 nights. She spent twice as many nights in Dallas as she did in Washington. The total
Cost for 22 nights (excluding tax) was $3,100. Determine the number of nights that she spent in each city. $$\begin{array} { l | c } { \text { City } } & \text { Cost per Night } \\\hline \text { Washington } & \$ 100 \\\hline \text { Atlanta } & \$ 175 \\\hline \text { Dallas } & \$ 150\end{array}$$

A) 5 nights in Washington, 7 nights in Atlanta, and 10 nights in Dallas
B) 1 night in Washington, 19 nights in Atlanta, and 2 nights in Dallas
C) 4 nights in Washington, 10 nights in Atlanta, and 8 nights in Dallas
D) 6 nights in Washington, 4 nights in Atlanta, and 12 nights in Dallas

Question 33

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{aligned}5 x - 7 y & = - 3 \\- 6 x + 5 y & = 7\end{aligned}$$

A) $\{ ( - 2 , - 1 ) \}$
B) $\{ ( - 1 , - 2 ) \}$
C) $\{ ( 2,1 ) \}$
D) $\{ ( - 1,2 ) \}$

Question 34

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
True or false? A system of linear equations in three variables may have exactly one solution.

Question 35

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
True or false? A system of linear equations in three variables may have no solution.

Question 36

Solve the problem.
Andre borrowed $40,000 to buy a truck for his business. He borrowed from his parents who charge him 3% simple interest. He borrowed from a credit union that charges 4% simple interest, and he
Borrowed from a bank that charges 6% simple interest. He borrowed four times as much from his
Parents as from the bank, and the amount of interest he paid at the end of 1 yr was $1,560. How
Much did he borrow from each source?

A) He borrowed $4,000 from his parents, $35,000 from the credit union, and $1,000 from the bank.
B) He borrowed 1,500 from his parents, 32,500 from the credit union, and 6,000 from the bank.
C) He borrowed 6,000 from his parents, 32,500 from the credit union, and 1,500 from the bank.
D) He borrowed $8,000 from his parents, $30,000 from the credit union, and $2,000 from the bank.

Question 37

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{aligned}x + 4 y & = - 7 \\4 x + 3 y & = 11\end{aligned}$$

A) $\{ ( 5 , - 3 ) \}$
В) $\left\{ \left( \frac { 17 } { 19 } , \frac { 65 } { 19 } \right) \right\}$
C) $\{ ( - 3,5 ) \}$
D) $\left\{ \left\{ - \frac { 65 } { 19 } , - \frac { 17 } { 19 } \right) \right\}$

Question 38

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } x _ { 1 } - 5 x _ { 2 } + 2 x _ { 3 } = 6 \\5 x _ { 2 } - 2 x _ { 3 } = - 1 \\- 3 x _ { 2 } - 4 x _ { 3 } - 5 x _ { 4 } = - 30 \\5 x _ { 1 } + 4 x _ { 2 } + 4 x _ { 3 } + 5 x _ { 4 } = 56\end{array}$$

A) $\{ ( 5,1 , - 3 , - 3 ) \}$
В) $\{ ( - 5 , - 1 , - 3 , - 3 ) \}$
C) $\{ ( - 5 , - 1,3,3 ) \}$
D) $\{ ( 5,1,3,3 ) \}$

Question 39

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } w - 2 x + 3 y = 8 \\- 5 x - 5 y = - 25 \\- 2 x + 3 y - 4 z = 17 \\- 3 w - 5 x + 2 y - 3 z = - 4\end{array}$$

A) $\{ ( - 3 , - 2,3 , - 3 ) \}$
B) $\{ ( - 3 , - 2 , - 3,3 ) \}$
C) $\{ ( 3,2,3 , - 3 ) \}$
D) $\{ ( 3,2 , - 3,3 ) \}$

Question 40

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } - 5 x + 9 y - 9 z = - 106 \\2 x - 2 y - 5 z = - 14 \\- 5 x - 8 y + 2 z = 11\end{array}$$

A) $\{ ( - 3 , - 5 , - 6 ) \}$
В) $\{ ( - 3,5,6 ) \}$
C) $\{ ( - 5,3,6 ) \}$
D) $\{ ( 5 , - 3,6 ) \}$

Question 41

Choose the one alternative that best completes the statement or answers the question.
For the given augmented matrix, determine the number of solutions to the corresponding system of
equations.
$$\left[ \begin{array} { l l l | r } 1 & 0 & 0 & - 3 \\0 & 1 & 0 & - 9 \\0 & 0 & 1 & 0\end{array} \right]$$

A) No solution
B) One solution
C) Infinitely many solutions

Question 42

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
If a system of linear equations has infinitely many solutions, then the equations are said to be
.

Question 43

The solution set to a system of dependent equations is given. Write the specific solution corresponding to
the given value of z.
$\{ ( 4 z - 7 , z - 4 , z ) \mid z$ is any real number $\}$ $z = - 5$

A) $( - 27,1 , - 5 )$
В) $( - 27 , - 9 , - 5 )$
C) $( - 13,1 , - 5 )$
D) $( - 13 , - 9 , - 5 )$

Question 44

Determine the solution set for the system represented by the augmented matrix.
$\left[ \begin{array} { r r r | r } 1 & 0 & - 7 & 4 \\ 0 & 1 & 6 & 1 \\ 0 & 0 & 0 & 1 \end{array} \right]$

A) $\{ ( - 7,6,0 ) \}$
В) $\{ ( 11 , - 5,1 ) \}$
C) $\{ ( 4,1,0 ) \}$
D){ }

Question 45

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
Assume that traffic flows freely through intersections A, B, and C. The values $$x _ { 1 } , x _ { 2 } , x _ { 3 }$$ , and all other
numbers in the figure represent flow rates in vehicles per hour. a. Write an equation representing equal flow into and out of intersection A.
b. Write an equation representing equal flow into and out of intersection B.
c. Write an equation representing equal flow into and out of intersection C.
d. Write the system of equations from parts (a)-(c) in standard form.
e. Write the reduced row-echelon form of the augmented matrix representing the system of equations from
part (d).
f. If the flow rate between intersections A and C is 120 vehicles per hour, determine the flow rates $x _ { 1 }$ and g. If the flow rate between intersections A and C is between 150 and 240 vehicles per hour, inclusive,
determine the flow rates

Question 46

The solution set to a system of dependent equations is given. Write three ordered triples that are
solutions to the system. Answers may vary.
$\left\{ \left( \frac { 6 - 4 y - 15 z } { 3 } , y , z \right) \mid y \right.$ and $z$ are any real numbers $\}$

A) $( 5,0,0 ) , ( 2,3,0 ) , ( - 3,0,1 )$
B) $( 2,0,0 ) , ( 2,3,0 ) , ( 1,0,1 )$
C) $( 5,0,0 ) , ( - 2,3,0 ) , ( 1,0,1 )$
D) $( 2,0,0 ) , ( - 2,3,0 ) , ( - 3,0,1 )$

Question 47

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
If a system of linear equations has no solution, then the system is said to be .

Question 48

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } - 6 x + 15 y = 15 \\2 x - 5 y = - 5\end{array}$$

A) $\{ ( 15 , - 5 ) \}$
B) $\left\{ \left( \frac { 5 } { 2 } x - \frac { 5 } { 2 } , y \right) \mid y \right.$ is any real number
C) $\{ ( - 9 , - 5 ) \}$
D){}

Question 49

Choose the one alternative that best completes the statement or answers the question.
For the given augmented matrix, determine the number of solutions to the corresponding system of
equations.
$$\left[ \begin{array} { r r r | r } 1 & 0 & 7 & - 2 \\0 & 1 & - 8 & 1 \\0 & 0 & 0 & 0\end{array} \right]$$

A) One solution
B) Infinitely many solutions
C) No solution

Question 50

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } 8 x + 10 y = - 17 \\4 x + 5 y = - 9\end{array}$$

A) $\{ ( - 7,7 ) \}$
B) $\{ ( - 17 , - 9 ) \}$
C) $\left\{ \left( - \frac { 5 } { 4 } x - \frac { 9 } { 4 } , y \right) \mid y \right.$ is any real number $\}$
D) { }

Question 51

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } 3 x + 2 y + 7 z = 38 \\9 x + 6 y + 21 z = 114 \\0.6 x + 0.4 y + 1.4 z = 7.6 \\\begin{array} { l l } \text { A) } \{ \} & \text { B) } \{ ( x , y , z ) \mid 3 x + 2 y + 7 z = 38 \} \\\text { C) } \{ ( - 1 , - 4,7 \} ) & \text { D) } \{ ( 1,4 , - 7 ) \}\end{array}\end{array}$$

Question 52

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } - 4 x - 6 y + 5 z = 13 \\8 x + 4 y - 7 z = - 7 \\4 x - 2 y - 2 z = - 15\end{array}$$

A) $\{ ( 8,4,5 ) \}$
В) $\{ ( 8,5,4 ) \}$
C) { }
D) $\{ ( 5,8,4 ) \}$

Question 53

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } - 2 x - y + 3 z = 13 \\x - 3 y - 3 z = - 4\end{array}$$

A) $\{ ( - 1 , - 2,3 ) \}$
B) $\left\{ \left( \frac { 12 } { 7 } z - \frac { 43 } { 7 } , - \frac { 3 } { 7 } z - \frac { 5 } { 7 } , z \right) \mid z \right.$ is any real number $\}$
C){}
D) $\{ ( 1,2 , - 3 ) \}$

Question 54

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.

Question 55

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
True or false? A system of linear equations in three variables may have infinitely many solutions.

Question 56

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
If the of a matrix is , then p represents the number of and q
represents the number of .

Question 57

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.

-An accountant checks the reported earnings for a concert venue for three different performers against the number of tickets sold. $\begin{array}{ccccc}\text { Performer } & \begin{array}{c}\text { Children } \\\text { Tickets }\end{array} & \begin{array}{c}\text { Student } \\\text { Tickets }\end{array} & \begin{array}{c}\text { General } \\\text { Admission }\end{array} & \begin{array}{c}\text { Total } \\\text { Revenue }\end{array} \\\mathbf{1} & 1,000 & 250 & 1,000 & \$ 24,750 \\\mathbf{2} & 400 & 300 & 300 & \$ 10,700 \\\mathbf{3} & 400 & 300 & 300 & \$ 8,700\end{array}$
Let $x , y$ , and $z$ represent the cost for children tickets, student tickets, and general admission tickets, respectively. Set up an augmented matrix for the system and solve for $x , y$ , and $z$ . Explain what the accountant knows about the reported earnings.

A){ } ; The system of equations reduces to a contradiction. There are no values for $x , y$ , and $z$ that can simultaneously meet the conditions of this problem.
B) $\{ ( 7,7,16 ) \}$ ; There were $\$ 7$ children's tickets, $\$ 7$ student tickets, and $\$ 16$ general admission tickets sold.
C) $\{ ( 8,8,17 ) \}$ ; There were $\$ 8$ children's tickets, $\$ 8$ student tickets, and $\$ 17$ general admission tickets sold.
D) $\{ ( 6,6,15 ) \}$ ; There were $\$ 6$ children's tickets, $\$ 6$ student tickets, and $\$ 15$ general admission tickets sold.

Question 58

Determine the solution set for the system represented by the augmented matrix.
$\left[ \begin{array} { r r r | r } 1 & 0 & - 4 & 7 \\ 0 & 1 & 8 & - 7 \\ 0 & 0 & 0 & 0 \end{array} \right]$

A) $\{ ( 7 - 4 z , - 7 + 8 z , z ) \mid z$ is any real number $\}$
B){ }
C) $\{ ( 7 , - 7,0 ) \}$
D) $\{ ( 7 + 4 z , - 7 - 8 z , z ) \mid z$ is any real number $\}$

Question 59

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } - 3 x - 3 y - 3 z = 30 \\- 9 x - 9 y - 9 z = 90 \\- 1.5 x - 1.5 y - 1.5 z = 15 \\\begin{array} { l l } \text { A) } \{ \} & \text { B) } \{ ( 2,2,6 ) \} \\\text { C) } \{ ( - 2 , - 2 , - 6 \} ) & \text { D) } \{ ( x , y , z ) \mid - 3 x - 3 y - 3 z = 30 \}\end{array}\end{array}$$

Question 60

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } - 3 x - 7 y + 7 z = 71 \\- 2 x + 7 y - 3 z = - 22 \\- 3 x - 9 y + 9 z = 87\end{array}$$

A) $\{ ( - 5 , - 2,6 ) \}$
B){ }
C) $\{ ( 5,2,6 ) \}$
D) $\{ ( - 5,2,6 ) \}$

Question 61

Give the order of the matrix. Classify the matrix as a square matrix, row matrix, column matrix, or none
of these.

Question 62

Find A + B.

Question 63

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
A matrix with the same number of rows and columns is called a matrix.

Question 64

For what values of x, y, and z will A = B?

Question 65

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
An matrix whose elements are all zero is called a matrix.

Question 66

Determine the value of the given element of the matrix.

Question 67

Find B - A.

Question 68

Give the order of the matrix. Classify the matrix as a square matrix, row matrix, column matrix, or none
of these.

Question 69

Find C - A + B

Question 70

Choose the one alternative that best completes the statement or answers the question.
Give the order of the matrix.

Question 71

Give the order of the matrix. Classify the matrix as a square matrix, row matrix, column matrix, or none
of these.

Question 72

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
True or false: If a row matrix A and a column matrix B have the same number of elements, then the
product AB is well defined.

Question 73

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
To multiply two matrices A and B, the number of of A must equal the number of
of B.

Question 74

Find A + B.

Question 75

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
True or false: Matrix multiplication is a commutative operation.

Question 76

Classify the matrix as a square matrix, row matrix, column matrix, or none of these.

Question 77

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
What are the requirements for two matrices to be equal?

Question 78

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
If A is a 3 matrix and B is a trix, then the product AB will be a matrix of order
. The product BA (is/is not) defined.

Question 79

Give the order of the matrix. Classify the matrix as a square matrix, row matrix, column matrix, or none
of these.

Question 80

Find the additive inverse of A.