In Exercises 19-22, determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. 19. 0 3 1 h 6 4 3] 8 20. 1 -2 h -3 4 6
In Exercises 19-22, determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. 19. 0 3 1 h 6 4 3] 8 20. 1 -2 h -3 4 6
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
19
![1
10 CHAPTER 1 Linear Equations in Linear Algebra
1.1 EXERCISES
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.
1.
=
*1 + 5^2 = 7
-2x₁ - 7x₂ = −5
2x₁ + 4x2 = −4
5x₁ + 7x₂ = 11
3. Find the point (x₁, x₂) that lies on the line x₁ + 5x₂ = 7 and
on the line x₁2x2 = -2. See the figure.
x2
x₁ - 2x₂ = -2
*+5^2 = 7
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.
5.
2017
OVERH
4. Find the point of intersection of the lines x₁ - 5x₂ = 1 and
3x₁ - 7x₂ = 5.
6.
7.
9.
10.
11.
1 -4 5 0
0
1 -3 0
0
0
1
0
2
0
0
0 1-5
DS
1 -6
0
0
0
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.
0
1
7 3 -4
0 1 -1
0
0
0
1
0
0
0
0
0
0
NE
sd bloode
4
0 -1
2 -7
0
4
0
1
2 -3
0
3 1 6
-1
1-2
7-
0
o owo - a
1 -3
0
1
0
0
7
0 -4
0 -7
-3 -1
002 4
3
1
1 -2
6asse
2.
3 -2
0-4 7
1 0 6
0 1-3
8.
Solve the systems in Exercises 11-14.
x₂ + 4x3 = -5
x₁ + 3x₂ + 5x3 = -2
3x₁ + 7x₂ + 7x3 =
6
001
-4
1
0 1
0
9
7
2
0
0
0
12.
13.
14.
X₁ - 3x₂ + 4x3 = -4
3x₁ - 7x2 + 7x3 = -8
-4x₁ + 6x₂ - X3 =
7
16.
Determine if the systems in Exercises 15 and 16 are consistent.
Do not completely solve the systems.
15.
+ 3x3
XI
- 3x3 = 8
2x1 + 2x2 + 9x3 = 7
X₂ + 5x3 = -2
X1 - 3x₂
= 5
-x₁ + x₂ + 5x3 = 2
X₂ + x3 = 0
19.
X1
21.
3x1
XI
- 3x4 =
X2
- 2x2 + 3x3 + 2x4 =
- 2x4 = -3
0
X3 + 3x4 =
1
X4
-2x1 + 3x2 + 2x3 + x4 = 5
2x2 + 2x3
17. Do the three lines x₁ - 4x2 = 1, 2x1 - x2 = -3, and
-x₁3x₂ = 4 have a common point of intersection?
Explain.
1
[3
3
18. Do the three planes x₁ + 2x₂ + x3 = 4, X₂ X3 = 1, and
x₁ + 3x2 = 0 have at least one common point of intersec-
tion? Explain.
3
1
+ 7x4 = -5
In Exercises 19-22, determine the value(s) of h such that the
matrix is the augmented matrix of a consistent linear system.
h
= 2
6
1
-4 h
4
8]
3-2
=
-3]
8
20.
22.
1
[-2
[-_3²3
-
h -3
4
-3]
6
-6 9
2 -3 h
1/2]
5
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
Ishows the statement is not true in all cases.) Similar true/false
questions will appear in many sections of the text.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87eb798c-8c88-4a7f-a953-6e87287b07b1%2Fdd8a997f-3bcc-459d-a983-f2b2ae46df85%2F44tsel_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1
10 CHAPTER 1 Linear Equations in Linear Algebra
1.1 EXERCISES
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.
1.
=
*1 + 5^2 = 7
-2x₁ - 7x₂ = −5
2x₁ + 4x2 = −4
5x₁ + 7x₂ = 11
3. Find the point (x₁, x₂) that lies on the line x₁ + 5x₂ = 7 and
on the line x₁2x2 = -2. See the figure.
x2
x₁ - 2x₂ = -2
*+5^2 = 7
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.
5.
2017
OVERH
4. Find the point of intersection of the lines x₁ - 5x₂ = 1 and
3x₁ - 7x₂ = 5.
6.
7.
9.
10.
11.
1 -4 5 0
0
1 -3 0
0
0
1
0
2
0
0
0 1-5
DS
1 -6
0
0
0
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.
0
1
7 3 -4
0 1 -1
0
0
0
1
0
0
0
0
0
0
NE
sd bloode
4
0 -1
2 -7
0
4
0
1
2 -3
0
3 1 6
-1
1-2
7-
0
o owo - a
1 -3
0
1
0
0
7
0 -4
0 -7
-3 -1
002 4
3
1
1 -2
6asse
2.
3 -2
0-4 7
1 0 6
0 1-3
8.
Solve the systems in Exercises 11-14.
x₂ + 4x3 = -5
x₁ + 3x₂ + 5x3 = -2
3x₁ + 7x₂ + 7x3 =
6
001
-4
1
0 1
0
9
7
2
0
0
0
12.
13.
14.
X₁ - 3x₂ + 4x3 = -4
3x₁ - 7x2 + 7x3 = -8
-4x₁ + 6x₂ - X3 =
7
16.
Determine if the systems in Exercises 15 and 16 are consistent.
Do not completely solve the systems.
15.
+ 3x3
XI
- 3x3 = 8
2x1 + 2x2 + 9x3 = 7
X₂ + 5x3 = -2
X1 - 3x₂
= 5
-x₁ + x₂ + 5x3 = 2
X₂ + x3 = 0
19.
X1
21.
3x1
XI
- 3x4 =
X2
- 2x2 + 3x3 + 2x4 =
- 2x4 = -3
0
X3 + 3x4 =
1
X4
-2x1 + 3x2 + 2x3 + x4 = 5
2x2 + 2x3
17. Do the three lines x₁ - 4x2 = 1, 2x1 - x2 = -3, and
-x₁3x₂ = 4 have a common point of intersection?
Explain.
1
[3
3
18. Do the three planes x₁ + 2x₂ + x3 = 4, X₂ X3 = 1, and
x₁ + 3x2 = 0 have at least one common point of intersec-
tion? Explain.
3
1
+ 7x4 = -5
In Exercises 19-22, determine the value(s) of h such that the
matrix is the augmented matrix of a consistent linear system.
h
= 2
6
1
-4 h
4
8]
3-2
=
-3]
8
20.
22.
1
[-2
[-_3²3
-
h -3
4
-3]
6
-6 9
2 -3 h
1/2]
5
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
Ishows the statement is not true in all cases.) Similar true/false
questions will appear in many sections of the text.
Expert Solution
Step 1
19.
Given Augmented matrix is
Apply the row operator then we get
For given system consistency we have
where = Number of non zero rows in Echlon form of Augmented matrix = 2 ( )
It follows that
This happen only if
So if is any real number then the given system is consistent and it has unique solution.
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