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Linear Algebra and Its Applications

David C. Lay, Steven R. Lay, Judi J. McDonald

Chapter 1

Linear Equations in Linear Algebra - all with Video Answers

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Section 1

Systems of Linear Equations

02:57

Problem 1

Solve each system in Exercises $1-4$ by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section.
$$
\begin{array}{rr}{x_{1}+5 x_{2}=} & {7} \\ {-2 x_{1}-7 x_{2}=} & {-5}\end{array}
$$

Willis James
Willis James
Numerade Educator
01:39

Problem 2

Solve each system in Exercises $1-4$ by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section.
$$
\begin{array}{l}{2 x_{1}+4 x_{2}=-4} \\ {5 x_{1}+7 x_{2}=11}\end{array}
$$

Breanna Ollech
Breanna Ollech
Numerade Educator
01:21

Problem 3

Solve each system in Exercises $1-4$ by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section.
Find the point $\left(x_{1}, x_{2}\right)$ that lies on the line $x_{1}+5 x_{2}=7$ and on the line $x_{1}-2 x_{2}=-2 .$ See the figure.

Samantha Lincroft
Samantha Lincroft
Numerade Educator
02:04

Problem 4

Solve each system in Exercises $1-4$ by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section.
Find the point of intersection of the lines $x_{1}-5 x_{2}=1$ and $3 x_{1}-7 x_{2}=5$

Alina Krajewski
Alina Krajewski
Numerade Educator
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Problem 5

Consider each matrix in Exercises 5 and 6 as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system.
$$
\left[\begin{array}{rrrrr}{1} & {-4} & {5} & {0} & {7} \\ {0} & {1} & {-3} & {0} & {6} \\ {0} & {0} & {1} & {0} & {2} \\ {0} & {0} & {0} & {1} & {-5}\end{array}\right]
$$

James Kiss
James Kiss
Numerade Educator
02:43

Problem 6

Consider each matrix in Exercises 5 and 6 as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system.
$$
\left[\begin{array}{rrrrr}{1} & {-6} & {4} & {0} & {-1} \\ {0} & {2} & {-7} & {0} & {4} \\ {0} & {0} & {1} & {2} & {-3} \\ {0} & {0} & {3} & {1} & {6}\end{array}\right]
$$

Alina Krajewski
Alina Krajewski
Numerade Educator
View

Problem 7

In Exercises 7–10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system.
$$
\left[\begin{array}{rrrr}{1} & {7} & {3} & {-4} \\ {0} & {1} & {-1} & {3} \\ {0} & {0} & {0} & {1} \\ {0} & {0} & {1} & {-2}\end{array}\right]
$$

James Kiss
James Kiss
Numerade Educator
02:41

Problem 8

In Exercises 7–10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system.
$$
\left[\begin{array}{rrrr}{1} & {-4} & {9} & {0} \\ {0} & {1} & {7} & {0} \\ {0} & {0} & {2} & {0}\end{array}\right]
$$

Alina Krajewski
Alina Krajewski
Numerade Educator
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Problem 9

In Exercises 7–10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system.
$$
\left[\begin{array}{rrrrr}{1} & {-1} & {0} & {0} & {-4} \\ {0} & {1} & {-3} & {0} & {-7} \\ {0} & {0} & {1} & {-3} & {-1} \\ {0} & {0} & {0} & {2} & {4}\end{array}\right]
$$

James Kiss
James Kiss
Numerade Educator
02:59

Problem 10

In Exercises 7–10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system.
$$
\left[\begin{array}{rrrrr}{1} & {-2} & {0} & {3} & {-2} \\ {0} & {1} & {0} & {-4} & {7} \\ {0} & {0} & {1} & {0} & {6} \\ {0} & {0} & {0} & {1} & {-3}\end{array}\right]
$$

Alina Krajewski
Alina Krajewski
Numerade Educator
02:05

Problem 11

Solve the systems in Exercises $11-14$
$$
\begin{aligned} x_{2}+4 x_{3} &=-5 \\ x_{1}+3 x_{2}+5 x_{3} &=-2 \\ 3 x_{1}+7 x_{2}+7 x_{3} &=6 \end{aligned}
$$

Sarah Gift
Sarah Gift
Numerade Educator
02:57

Problem 12

Solve the systems in Exercises $11-14$
$$
\begin{aligned} x_{1}-3 x_{2}+4 x_{3} &=-4 \\ 3 x_{1}-7 x_{2}+7 x_{3} &=-8 \\-4 x_{1}+6 x_{2}-x_{3} &=7 \end{aligned}
$$

Alina Krajewski
Alina Krajewski
Numerade Educator
05:22

Problem 13

Solve the systems in Exercises $11-14$
$$
\begin{aligned} x_{1} &-3 x_{3}=8 \\ 2 x_{1}+2 x_{2}+9 x_{3} &=7 \\ x_{2}+5 x_{3} &=-2 \end{aligned}
$$

Samantha Lincroft
Samantha Lincroft
Numerade Educator
03:50

Problem 14

Solve the systems in Exercises $11-14$
$$
\begin{aligned} x_{1}-3 x_{2} &=5 \\-x_{1}+x_{2}+5 x_{3} &=2 \\ x_{2}+x_{3} &=0 \end{aligned}
$$

H M
H M
Numerade Educator
02:47

Problem 15

Determine if the systems in Exercises 15 and 16 are consistent. Do not completely solve the systems.
$$
\begin{aligned} x_{1} &+3 x_{3} \quad=2 \\ x_{2} &-3 x_{4}=3 \\-2 x_{2}+3 x_{3}+2 x_{4} &=1 \\ 3 x_{1} &+7 x_{4}=-5 \end{aligned}
$$

WM
William Mead
Numerade Educator
03:01

Problem 16

Determine if the systems in Exercises 15 and 16 are consistent. Do not completely solve the systems.
$$
\begin{aligned} x_{1} &-2 x_{4}=-3 \\ 2 x_{2}+2 x_{3} &=0 \\ x_{3}+3 x_{4} &=1 \\-2 x_{1}+3 x_{2}+2 x_{3}+x_{4} &=5 \end{aligned}
$$

Alina Krajewski
Alina Krajewski
Numerade Educator
View

Problem 17

Do the three lines $x_{1}-4 x_{2}=1,2 x_{1}-x_{2}=-3,$ and $-x_{1}-3 x_{2}=4$ have a common point of intersection? Explain.

Danielle Fairburn
Danielle Fairburn
Numerade Educator
01:40

Problem 18

Do the three planes $x_{1}+2 x_{2}+x_{3}=4, x_{2}-x_{3}=1,$ and $x_{1}+3 x_{2}=0$ have at least one common point of intersection? Explain.

Alina Krajewski
Alina Krajewski
Numerade Educator
00:59

Problem 19

In Exercises $19-22$ , determine the value(s) of $h$ such that the matrix is the augmented matrix of a consistent linear system.
$$
\left[\begin{array}{lll}{1} & {h} & {4} \\ {3} & {6} & {8}\end{array}\right]
$$

Samantha Lincroft
Samantha Lincroft
Numerade Educator
01:35

Problem 20

In Exercises $19-22$ , determine the value(s) of $h$ such that the matrix is the augmented matrix of a consistent linear system.
$$
\left[\begin{array}{rrr}{1} & {h} & {-3} \\ {-2} & {4} & {6}\end{array}\right]
$$

Alina Krajewski
Alina Krajewski
Numerade Educator
00:54

Problem 21

In Exercises $19-22$ , determine the value(s) of $h$ such that the matrix is the augmented matrix of a consistent linear system.
$$
\left[\begin{array}{rrr}{1} & {3} & {-2} \\ {-4} & {h} & {8}\end{array}\right]
$$

Samantha Lincroft
Samantha Lincroft
Numerade Educator
01:11

Problem 22

In Exercises $19-22$ , determine the value(s) of $h$ such that the matrix is the augmented matrix of a consistent linear system.
$$
\left[\begin{array}{rrr}{2} & {-3} & {h} \\ {-6} & {9} & {5}\end{array}\right]
$$

Alina Krajewski
Alina Krajewski
Numerade Educator
02:58

Problem 23

In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or refer to a definition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text.
a. Every elementary row operation is reversible.
b. A $5 \times 6$ matrix has six rows.
c. The solution set of a linear system involving variables $x_{1}, \ldots, x_{n}$ is a list of numbers $\left(s_{1}, \ldots, s_{n}\right)$ that makes each equation in the system a true statement when the values $s_{1}, \ldots, s_{n}$ are substituted for $x_{1}, \ldots, x_{n},$ respectively.
d. Two fundamental questions about a linear system involve existence and uniqueness.

Samantha Lincroft
Samantha Lincroft
Numerade Educator
02:08

Problem 24

In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or refer to a definition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text.
a. Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
b. Two matrices are row equivalent if they have the same number of rows.
c. An inconsistent system has more than one solution.
d. Two linear systems are equivalent if they have the same solution set.

Alina Krajewski
Alina Krajewski
Numerade Educator
01:24

Problem 25

Find an equation involving $g, h,$ and $k$ that makes this augmented matrix correspond to a consistent system:
$$
\left[\begin{array}{rrrr}{1} & {-4} & {7} & {g} \\ {0} & {3} & {-5} & {h} \\ {-2} & {5} & {-9} & {k}\end{array}\right]
$$

Samantha Lincroft
Samantha Lincroft
Numerade Educator
02:36

Problem 26

Construct three different augmented matrices for linear systems whose solution set is $x_{1}=-2, x_{2}=1, x_{3}=0$.

Alina Krajewski
Alina Krajewski
Numerade Educator
00:42

Problem 27

Suppose the system below is consistent for all possible values of $f$ and $g .$ What can you say about the coefficients $c$ and $d ?$ Justify your answer.
$$\begin{aligned} x_{1}+3 x_{2} &=f \\ c x_{1}+d x_{2} &=g \end{aligned}$$

Samantha Lincroft
Samantha Lincroft
Numerade Educator
02:33

Problem 28

Suppose $a, b, c,$ and $d$ are constants such that $a$ is not zero and the system below is consistent for all possible values of $f$ and $g .$ What can you say about the numbers $a, b, c,$ and $d ?$ Justify your answer.
$$
\begin{array}{l}{a x_{1}+b x_{2}=f} \\ {c x_{1}+d x_{2}=g}\end{array}
$$

Alina Krajewski
Alina Krajewski
Numerade Educator
01:31

Problem 29

In Exercises 29–32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.
$$
\left[\begin{array}{rrr}{0} & {-2} & {5} \\ {1} & {4} & {-7} \\ {3} & {-1} & {6}\end{array}\right],\left[\begin{array}{rrr}{1} & {4} & {-7} \\ {0} & {-2} & {5} \\ {3} & {-1} & {6}\end{array}\right]
$$

WM
William Mead
Numerade Educator
01:29

Problem 30

In Exercises 29–32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.
$$
\left[\begin{array}{rrr}{1} & {3} & {-4} \\ {0} & {-2} & {6} \\ {0} & {-5} & {9}\end{array}\right],\left[\begin{array}{rrr}{1} & {3} & {-4} \\ {0} & {1} & {-3} \\ {0} & {-5} & {9}\end{array}\right]
$$

Alina Krajewski
Alina Krajewski
Numerade Educator
01:19

Problem 31

In Exercises 29–32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.
$$
\left[\begin{array}{rrrr}{1} & {-2} & {1} & {0} \\ {0} & {5} & {-2} & {8} \\ {4} & {-1} & {3} & {-6}\end{array}\right],\left[\begin{array}{rrrr}{1} & {-2} & {1} & {0} \\ {0} & {5} & {-2} & {8} \\ {0} & {7} & {-1} & {-6}\end{array}\right]
$$

Samantha Lincroft
Samantha Lincroft
Numerade Educator
01:55

Problem 32

In Exercises 29–32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.
$$
\left[\begin{array}{rrrr}{1} & {2} & {-5} & {0} \\ {0} & {1} & {-3} & {-2} \\ {0} & {-3} & {9} & {5}\end{array}\right],\left[\begin{array}{rrrr}{1} & {2} & {-5} & {0} \\ {0} & {1} & {-3} & {-2} \\ {0} & {0} & {0} & {-1}\end{array}\right]
$$

Alina Krajewski
Alina Krajewski
Numerade Educator
01:05

Problem 33

An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let $T_{1}, \ldots, T_{4}$ denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes - to the left, above, to the right, and below.' For instance,
$$
T_{1}=\left(10+20+T_{2}+T_{4}\right) / 4, \quad \text { or } \quad 4 T_{1}-T_{2}-T_{4}=30
$$
(PICTURE NOT COPY)
Write a system of four equations whose solution gives estimates for the temperatures $T_{1}, \ldots, T_{4}$

Samantha Lincroft
Samantha Lincroft
Numerade Educator
18:21

Problem 34

An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let $T_{1}, \ldots, T_{4}$ denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodesto the left, above, to the right, and below. ${ }^{2}$ For instance,
$T_{1}=\left(10+20+T_{2}+T_{4}\right) / 4, \quad$ or $\quad 4 T_{1}-T_{2}-T_{4}=30$
Write a system of four equations whose solution gives estimates for the temperatures $T_{1}, \ldots, T_{4}$

Joseph Lentino
Joseph Lentino
Numerade Educator