- 0 0 . . -3 . 1 0 0 . ܝܚ 2 . 0 5 0 1 0 ܘ ܐ ܩ ܘ -2 1 4 . .
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
13
![22 CHAPTER 1 Linear Equations in Linear Algebra
3.
9.
1
2. a. 0
0
11.
13.
14.
C.
d.
15. a.
b.
1
4
6
1
0
0
0
-9
-6
1
1
0
0
3
0
0
0
0
3
0
1-2
0
0
Го
257
los Baodisab
Row reduce the matrices in Exercises 3 and 4 to reduced echelon
form. Circle the pivot positions in the final matrix and in the
original matrix, and list the pivot columns.
ETRY
4.
1
0
0
1
0 0
1-3
0
1
0
0
0
1
1
0
1
0
0
0
3 4
9 7
0
0
1
2
0
0
0
0 0
1 0
1
1
0
6
1 -6 5
7-6
3 4
6 7
7 8 9 Instavin
0
5. Describe the possible echelon forms of a nonzero 2 x 2
matrix. Use the symbols, *, and 0, as in the first part of
Example 1.
6. Repeat Exercise 5 for a nonzero 3 x 2 matrix.
Find the general solutions of the systems whose augmented ma-
trices are given in Exercises 7-14.
7.
-4 2 0
12 -6
1
8 -4 0
1
0
1
2 2
0 3
0 0
]
2 -5 -6
1 -6-3
0 0 01
0
*
*
*
*
■
0 ■ 0
*
00
0 000
*
b.
0 -1 0 -27
0
0-4 1
0
1 9 4
0 0 0
0-5
0 2
0
0 0 0 0
8.
10.
*
12.
1
0
0
HAS
[1
3
3
5
5 7
[2
[3
1
1
01
gas zoldsha
0
0
1
0
0
-1
1
4
7
1 -2 -1
-6 -2
0
0
1
5 7
7
9
9 1
Exercises 15 and 16 use the notation of Example 1 for matrices
in echelon form. Suppose each matrix represents the augmented
matrix for a system of linear equations. In each case, determine if
the system is consistent. If the system is consistent, determine if
the solution is unique.
71
10
07
2]
0 6
-7
5
0 1-2
-2 -3
7-4
2
7
16. a.
O
Wow
b.
17.
2
0
■
0
4
0
3
In Exercises 17 and 18, determine the value(s) of h such that the
matrix is the augmented matrix of a consistent linear system.
h
47]
6
:)
0 0
Jinsi In Exercises 19 and 20, choose h and k such that the system has
(a) no solution, (b) a unique solution, and (c) many solutions. Give
separate answers for each part.
0
00
19. x₁ + hx₂ = 2
Banom 4x₁ + 8x₂ = k
*
■
3
*
18.
1 -3 -2
5
h -7
20. x₁ + 3x₂ = 2
3x₁ + hx₂ = k
b. The row reduction algorithm applies only to augmented
is toxins in matrices for a linear system.
In Exercises 21 and 22, mark each statement True or False. Justify
each answer.4
21//a.
In some cases, a matrix may be row reduced to more
than one matrix in reduced echelon form, using different
sequences of row operations.
c. A basic variable in a linear system is a variable that
corresponds to a pivot column in the coefficient matrix.
d. Finding a parametric description of the solution set of a
22400 linear system is the same as solving the system.
lule
e. If one row in an echelon form of an augmented matrix
is [0 0 0 5 0], then the associated linear system is
inconsistent.
22. a The echelon form of a matrix is unique.
2
b. The pivot positions in a matrix depend on whether row
interchanges are used in the row reduction process.
c. Reducing a matrix to echelon form is called the forward
phase of the row reduction process.
d. Whenever a system has free variables, the solution set
contains many solutions.
e. A general solution of a system is an explicit description
of all solutions of the system.
23. Suppose a 3 x 5 coefficient matrix for a system has three
pivot columns. Is the system consistent? Why or why not?
24. Suppose a system of linear equations has a 3 x 5 augmented
matrix whose fifth column is a pivot column. Is the system
consistent? Why (or why not)?
4 True/false questions of this type will appear in many sections. Methods
for justifying your answers were described before Exercises 23 and 24 in
Section 1.1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3736a1e3-92be-43f4-a863-c20bb07961e7%2Fde6d8bd3-c74f-4267-a565-d742ae792216%2Fopiyv6iv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:22 CHAPTER 1 Linear Equations in Linear Algebra
3.
9.
1
2. a. 0
0
11.
13.
14.
C.
d.
15. a.
b.
1
4
6
1
0
0
0
-9
-6
1
1
0
0
3
0
0
0
0
3
0
1-2
0
0
Го
257
los Baodisab
Row reduce the matrices in Exercises 3 and 4 to reduced echelon
form. Circle the pivot positions in the final matrix and in the
original matrix, and list the pivot columns.
ETRY
4.
1
0
0
1
0 0
1-3
0
1
0
0
0
1
1
0
1
0
0
0
3 4
9 7
0
0
1
2
0
0
0
0 0
1 0
1
1
0
6
1 -6 5
7-6
3 4
6 7
7 8 9 Instavin
0
5. Describe the possible echelon forms of a nonzero 2 x 2
matrix. Use the symbols, *, and 0, as in the first part of
Example 1.
6. Repeat Exercise 5 for a nonzero 3 x 2 matrix.
Find the general solutions of the systems whose augmented ma-
trices are given in Exercises 7-14.
7.
-4 2 0
12 -6
1
8 -4 0
1
0
1
2 2
0 3
0 0
]
2 -5 -6
1 -6-3
0 0 01
0
*
*
*
*
■
0 ■ 0
*
00
0 000
*
b.
0 -1 0 -27
0
0-4 1
0
1 9 4
0 0 0
0-5
0 2
0
0 0 0 0
8.
10.
*
12.
1
0
0
HAS
[1
3
3
5
5 7
[2
[3
1
1
01
gas zoldsha
0
0
1
0
0
-1
1
4
7
1 -2 -1
-6 -2
0
0
1
5 7
7
9
9 1
Exercises 15 and 16 use the notation of Example 1 for matrices
in echelon form. Suppose each matrix represents the augmented
matrix for a system of linear equations. In each case, determine if
the system is consistent. If the system is consistent, determine if
the solution is unique.
71
10
07
2]
0 6
-7
5
0 1-2
-2 -3
7-4
2
7
16. a.
O
Wow
b.
17.
2
0
■
0
4
0
3
In Exercises 17 and 18, determine the value(s) of h such that the
matrix is the augmented matrix of a consistent linear system.
h
47]
6
:)
0 0
Jinsi In Exercises 19 and 20, choose h and k such that the system has
(a) no solution, (b) a unique solution, and (c) many solutions. Give
separate answers for each part.
0
00
19. x₁ + hx₂ = 2
Banom 4x₁ + 8x₂ = k
*
■
3
*
18.
1 -3 -2
5
h -7
20. x₁ + 3x₂ = 2
3x₁ + hx₂ = k
b. The row reduction algorithm applies only to augmented
is toxins in matrices for a linear system.
In Exercises 21 and 22, mark each statement True or False. Justify
each answer.4
21//a.
In some cases, a matrix may be row reduced to more
than one matrix in reduced echelon form, using different
sequences of row operations.
c. A basic variable in a linear system is a variable that
corresponds to a pivot column in the coefficient matrix.
d. Finding a parametric description of the solution set of a
22400 linear system is the same as solving the system.
lule
e. If one row in an echelon form of an augmented matrix
is [0 0 0 5 0], then the associated linear system is
inconsistent.
22. a The echelon form of a matrix is unique.
2
b. The pivot positions in a matrix depend on whether row
interchanges are used in the row reduction process.
c. Reducing a matrix to echelon form is called the forward
phase of the row reduction process.
d. Whenever a system has free variables, the solution set
contains many solutions.
e. A general solution of a system is an explicit description
of all solutions of the system.
23. Suppose a 3 x 5 coefficient matrix for a system has three
pivot columns. Is the system consistent? Why or why not?
24. Suppose a system of linear equations has a 3 x 5 augmented
matrix whose fifth column is a pivot column. Is the system
consistent? Why (or why not)?
4 True/false questions of this type will appear in many sections. Methods
for justifying your answers were described before Exercises 23 and 24 in
Section 1.1.
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