Precalculus Functions and Graphs

Mark Dugopolski

Chapter 9

Matrices and Determinants - all with Video Answers

Educators

Section 1

Solving Linear Systems Using Matrices

Problem 1

Fill in the blank.

A rectangular array of real numbers is a(n) ______ .

Xiaomeng Zhang

Xiaomeng Zhang

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Problem 2

Fill in the blank.

In a matrix, the ______ run horizontally and the ________ run vertically.

Xiaomeng Zhang

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Problem 3

Fill in the blank.

The ______ of a matrix is the number of rows and columns in the matrix.

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Problem 4

Fill in the blank.

A matrix with an equal number of rows and columns is a(n) ______ matrix.

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Problem 5

Fill in the blank.

Each number in a matrix is a(n) ______ or _______ of the matrix.

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Problem 6

Fill in the blank.

A matrix consisting of the coefficients and the constants from a system of equations is a(n) ______ matrix.

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Problem 7

Fill in the blank.

The _______ of a $2 \times 2$ matrix consists of the entries in the first row first column and the second row second column.

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Problem 8

Fill in the blank.

A square matrix with ones on the diagonal and zeros elsewhere is the ______ matrix.

Xiaomeng Zhang

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Problem 9

Determine the size of each matrix.

$\left[\begin{array}{lll}1 & 5 & 8\end{array}\right]$

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Problem 10

Determine the size of each matrix.

$\left[\begin{array}{r}-3 \\ y \\ 5\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 11

Determine the size of each matrix.

[ 7]

Xiaomeng Zhang

Numerade Educator

Problem 12

Determine the size of each matrix.

$\left[\begin{array}{ll}x & y \\ z & w\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 13

Determine the size of each matrix.

$\left[\begin{array}{rr}-5 & 12 \\ 99 & 6 \\ 0 & 0\end{array}\right]$

Xiaomeng Zhang

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Problem 14

Determine the size of each matrix.

$\left[\begin{array}{rrr}1 & 5 & 7 \\ 3 & 0 & 5 \\ 2 & -6 & -3\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 15

Write the augmented matrix for each system of equations.

$x-2 y=4$
$3 x+2 y=-5$

Xiaomeng Zhang

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Problem 16

Write the augmented matrix for each system of equations.

$4 x-y=1$
$x+3 y=5$

Xiaomeng Zhang

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Problem 17

Write the augmented matrix for each system of equations.

$\begin{aligned} x-y-z &=4 \\ x+3 y-z &=1 \\ 2 y-5 z &=-6 \end{aligned}$

Xiaomeng Zhang

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Problem 18

Write the augmented matrix for each system of equations.

$\begin{aligned} x+3 y &=5 \\ y-4 z &=8 \\-2 x &+5 z=7 \end{aligned}$

Xiaomeng Zhang

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Problem 19

Write the augmented matrix for each system of equations.

$x+3 y-z=5$
$x+z=0$

Xiaomeng Zhang

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Problem 20

Write the augmented matrix for each system of equations.

$x-y=6$
$x+z=7$

Xiaomeng Zhang

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Problem 21

Write the system of equations represented by each augmented matrix.

$\left[\begin{array}{rr|r}3 & 4 & -2 \\ 3 & -5 & 0\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 22

Write the system of equations represented by each augmented matrix.

$\left[\begin{array}{ll|r}1 & 0 & -7 \\ 0 & 1 & 5\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 23

Write the system of equations represented by each augmented matrix.

$\left[\begin{array}{rrr|r}5 & 0 & 0 & 6 \\ -4 & 0 & 2 & -1 \\ 4 & 4 & 0 & 7\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 24

Write the system of equations represented by each augmented matrix.

$\left[\begin{array}{rrr|r}1 & 0 & 1 & 2 \\ 0 & 1 & -1 & -6 \\ 1 & -1 & 1 & 5\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 25

Write the system of equations represented by each augmented matrix.

$\left[\begin{array}{rrr|r}1 & -1 & 2 & 1 \\ 0 & 1 & 4 & 3 \\ 0 & 0 & 0 & 0\end{array}\right]$

Xiaomeng Zhang

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Problem 26

Write the system of equations represented by each augmented matrix.

$\left[\begin{array}{lll|l}1 & 1 & 1 & 3 \\ 0 & 1 & 2 & 7 \\ 0 & 0 & 0 & 0\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 27

Perform the indicated row operation on the given augmented matrix. See the summary of row operations on page 626.

$R_{1} \leftrightarrow R_{2}:\left[\begin{array}{rr|r}-2 & 4 & 1 \\ 1 & 2 & 0\end{array}\right]$

Xiaomeng Zhang

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Problem 28

Perform the indicated row operation on the given augmented matrix. See the summary of row operations on page 626.

$R_{1} \leftrightarrow R_{2}:\left[\begin{array}{ll|l}4 & 9 & 3 \\ 1 & 0 & 5\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 29

Perform the indicated row operation on the given augmented matrix. See the summary of row operations on page 626.

$\frac{1}{2} R_{1} \rightarrow R_{1}:\left[\begin{array}{ll|l}2 & 8 & 2 \\ 0 & 3 & 6\end{array}\right]$

Xiaomeng Zhang

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Problem 30

Perform the indicated row operation on the given augmented matrix. See the summary of row operations on page 626.

$-\frac{1}{3} R_{1} \rightarrow R_{1}:\left[\begin{array}{rr|r}-3 & 6 & 12 \\ 0 & 9 & 3\end{array}\right]$

Xiaomeng Zhang

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Problem 31

Perform the indicated row operation on the given augmented matrix. See the summary of row operations on page 626.

$3 R_{1}+R_{2} \rightarrow R_{2}:\left[\begin{array}{rr|r}1 & -2 & 1 \\ -3 & 5 & 0\end{array}\right]$

Xiaomeng Zhang

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Problem 32

Perform the indicated row operation on the given augmented matrix. See the summary of row operations on page 626.

$-2 R_{2}+R_{1} \rightarrow R_{1}:\left[\begin{array}{ll|l}1 & 2 & 7 \\ 0 & 1 & 4\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 33

For each given sequence of augmented matrices determine the system that has been solved, the solution, and the row operation that was used on each matrix to obtain the next matrix in the sequence. For the row operations use the notation that was introduced in the examples.

$\left[\begin{array}{ll|r}2 & 4 & 14 \\ 5 & 4 & 5\end{array}\right]$
$\left[\begin{array}{ll|l}1 & 2 & 7 \\ 5 & 4 & 5\end{array}\right]$
$\left[\begin{array}{rr|r}1 & 2 & 7 \\ 0 & -6 & -30\end{array}\right]$
$\left[\begin{array}{ll|l}1 & 2 & 7 \\ 0 & 1 & 5\end{array}\right]$
$\left[\begin{array}{ll|r}1 & 0 & -3 \\ 0 & 1 & 5\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 34

For each given sequence of augmented matrices determine the system that has been solved, the solution, and the row operation that was used on each matrix to obtain the next matrix in the sequence. For the row operations use the notation that was introduced in the examples.

$\left[\begin{array}{ll|l}3 & 5 & -2 \\ 1 & 2 & -1\end{array}\right]$
$\left[\begin{array}{cc|c}1 & 2 & -1 \\ 3 & 5 & -2\end{array}\right]$
$\left[\begin{array}{rr|r}1 & 2 & -1 \\ 0 & -1 & 1\end{array}\right]$
$\left[\begin{array}{ll|l}1 & 2 & -1 \\ 0 & 1 & -1\end{array}\right]$
$\left[\begin{array}{ll|r}1 & 0 & 1 \\ 0 & 1 & -1\end{array}\right]$

Xiaomeng Zhang

Numerade Educator

Problem 35

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$\begin{aligned} x+y &=5 \\-2 x+y &=-1 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 36

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$\begin{aligned} x-y &=2 \\ 3 x-y &=12 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 37

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$\begin{aligned} 2 x+2 y &=8 \\-3 x-y &=-6 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 38

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$3 x-6 y=9$
$2 x+y=-4$

Xiaomeng Zhang

Numerade Educator

Problem 39

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$2 x-y=3$
$3 x+2 y=15$

Xiaomeng Zhang

Numerade Educator

Problem 40

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$2 x-3 y=-1$
$3 x-2 y=1$

Xiaomeng Zhang

Numerade Educator

Problem 41

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$\begin{aligned} 0.4 x-0.2 y &=0 \\ x+1.5 y &=2 \end{aligned}$

Xiaomeng Zhang

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Problem 42

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$0.2 x+0.6 y=0.7$
$0.5 x-y=0.5$

Xiaomeng Zhang

Numerade Educator

Problem 43

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$\begin{aligned} 3 a-5 b &=7 \\-3 a+5 b &=4 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 44

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$2 s-3 t=9$
$4 s-6 t=1$

Xiaomeng Zhang

Numerade Educator

Problem 45

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$\begin{aligned} 0.5 u+1.5 v &=2 \\ 3 u+9 v &=12 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 46

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$\begin{aligned} m-2.5 n &=0.5 \\-4 m+10 n &=-2 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 47

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$y=4-2 x$
$x=8+y$

Xiaomeng Zhang

Numerade Educator

Problem 48

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$3 x=1+2 y$
$y=2-x$

Xiaomeng Zhang

Numerade Educator

Problem 49

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$\begin{aligned} x+y+z &=6 \\ x-y-z &=0 \\ 2 y-z &=3 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 50

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$\begin{aligned} x-y+z &=2 \\-x+y+z &=4 \\-x &=2 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 51

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$2 x+y=2+z$
$x+2 y=2+z$
$x+2 z=2+y$

Xiaomeng Zhang

Numerade Educator

Problem 52

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$\begin{aligned} 3 x &=4+y \\ x+y &=z-1 \\ 2 z &=3-x \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 53

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$2 a-2 b+c=-2$
$a+b-3 c=3$
$a-3 b+c=-5$

Xiaomeng Zhang

Numerade Educator

Problem 54

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$\begin{aligned} r-3 s-t &=-3 \\-r-s+2 t &=1 \\-r+2 s-t &=-2 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 55

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$3 y=x+z$
$x-y-3 z=4$
$x+y+2 z=-1$

Xiaomeng Zhang

Numerade Educator

Problem 56

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$z=2+x$
$2 x-y=1$
$y+3 z=15$

Xiaomeng Zhang

Numerade Educator

Problem 57

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$\begin{aligned} x-2 y+3 z &=1 \\ 2 x-4 y+6 z &=2 \\-3 x+6 y-9 z &=-3 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 58

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$\begin{aligned} 4 x-2 y+6 z &=4 \\ 2 x-y+3 z &=2 \\-2 x+y-3 z &=-2 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 59

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$\begin{aligned} x-y+z &=2 \\ 2 x+y-z &=1 \\ 2 x-2 y+2 z &=5 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 60

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$x-y+z=4$
$x+y-z=1$
$x+y-z=3$

Xiaomeng Zhang

Numerade Educator

Problem 61

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$\begin{aligned} x+y-z &=3 \\ 3 x+y+z &=7 \\ x-y+3 z &=1 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 62

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$\begin{aligned} x+2 y+2 z &=4 \\ 2 x+y+z &=1 \\-x+y+z &=3 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 63

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$2 x-y+3 z=1$
$x+y-z=4$

Xiaomeng Zhang

Numerade Educator

Problem 64

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$x+3 y+z=6$
$-x+y-z=2$

Xiaomeng Zhang

Numerade Educator

Problem 65

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$\begin{aligned} x-y+z-w &=2 \\-x+2 y-z-w &=-1 \\ 2 x-y-z+w &=4 \\ x+3 y-2 z-3 w &=6 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 66

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for three equations on page 630.

$\begin{aligned} 3 a-2 b+c+d &=0 \\ a-b+c-d &=-4 \\-2 a+b+3 c-2 d &=5 \\ 2 a+3 b-c-d &=-3 \end{aligned}$

Xiaomeng Zhang

Numerade Educator

Problem 67

Wages from Two Jobs Mike works a total of 60 hr per week at his two jobs. He makes $\$ 8$ per hour at Burgers-R-Us and S9 per hour at the Soap Opera Laundromat. If his total pay for one week is $/$ 502$ before taxes, then how many hours does he work at each job?

Gregory Higby

Numerade Educator

Problem 68

Postal Rates Noriko spent $\$ 23.20$ on postage inviting a total of 60 guests to her promotion party. Each woman was sent a picture postcard showing the company headquarters in Tokyo while each man was invited with a letter. If she put a 28 -cent stamp on each postcard and a 44 -cent stamp on each letter, then how many guests of each gender were invited?

Xiaomeng Zhang

Numerade Educator

Problem 69

Investment Portfolio Petula invested a total of $\$ 40,000$ in a no-load mutual fund, treasury bills, and municipal bonds. Her total return of $\$ 3660$ came from an $8 \%$ return on her investment in the no-load mutual fund, a $9 \%$ return on the treasury bills, and $12 \%$ return on the municipal bonds. If her total investment in treasury bills and municipal bonds was equal to her investment in the no-load mutual fund, then how much did she invest in each?

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 70

Nutrition The accompanying table shows the percentage of U.S. Recommended Daily Allowances (RDA) for phosphorus (P), magnesium (Mg), and calcium (Ca) in one ounce of three breakfast cereals (without milk). If Hulk Hogan got $98 \%$ of the RDA of phosphorus, $84 \%$ of the RDA of magnesium, and $38 \%$ of the RDA of calcium by eating a large bowl of each (without milk), then how many ounces of each cereal did he eat?
Write an equation for $P$, one for Mg, and one for Ca.

(check your book to see table)

AG

Ankit Gupta

Numerade Educator

Problem 71

Cubic Curve Fitting Find $a, b,$ and $c$ such that the graph of $y=a x^{3}+b x+c$ goes through the points $(-1,4),(1,2)$ and $(2,7)$

Xiaomeng Zhang

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Problem 72

Quadratic Curve Fitting Find $a, b,$ and $c$ such that the graph of $y=a x^{2}+b x+c$ goes through the points $(-1,0),(1,0)$ and $(3,2)$

Xiaomeng Zhang

Numerade Educator

Problem 73

Traffic Control $I$ The diagram shows the number of cars that desire to enter and leave each of three intersections on three one-way streets in a 60-minute period. The letters $x, y$ and $z$ represent the number of cars passing the three points between these three intersections, as shown in the diagram. Find values for $x, y,$ and $z$ that would realize this desired traffic flow. If construction on JFK Boulevard limits the value of $z$ to $50,$ then what values for $x$ and $y$ would keep the traffic flowing?

Sherrie Fenner

Numerade Educator

Problem 74

Traffic Control II Southbound M. L. King Drive in the figure leads to another intersection that cannot always handle 700 cars in a 60 -minute period. Change 700 to 600 in the figure and write a system of three equations in $x, y,$ and $z .$ What is the solution to this system? If you had control over the 400 cars coming from the north into the first intersection on M. L. King Drive, what number would you use in place of 400 to get the system flowing again?

Ahmad Reda

Numerade Educator

Problem 75

Determine whether $(-2,1)$ satisfies the system of inequalities $x^{2}+y^{2}<9, y>x+1,$ and $x+y>2$

Xiaomeng Zhang

Numerade Educator

Problem 76

Solve the system $3 x-5 y=-5$ and $4 x+3 y=-26$.

Xiaomeng Zhang

Numerade Educator

Problem 77

Solve the system $x+y=4, y+z=-3,$ and $x+z=-17$.

Xiaomeng Zhang

Numerade Educator

Problem 78

Evaluate each expression.
a. $\log _{5}(1 / 25)$
b. $\log (\sqrt{10})$
c. $\ln \left(e^{3}\right)$

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Problem 79

Solve $|x-3|=|2 x+5|$

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Problem 80

Solve $15(2 x-1)^{2}-2(2 x-1)-8=0$

Xiaomeng Zhang

Numerade Educator