0-5 Fine find 25. 00-3 inding an Elementary Matrix In Exercises 9-12, t A, B, and C be 27. 2 Fine In and mat 29. 31. . Find an elementary matrix Fsuch that FA-R a. Find an elementary matris Fsuch that FA-r 1. Find an elementary matris Fsuch that FR-A 2 Find an elementary matrix F such that Fr 33. inding a Sequence of Elementary Matrices Exercises 13-18, find a sequence of elementary matrices at can be used to write the matrix in row-echelon form. 35 3 -3 14. -1 -2 37. 10 -5 2 I -2 -1 38. 8 -4 16. 2 3 5-1 12 -2 I 3 -4 I -2 -3 5 -1 4 8-5 18.
0-5 Fine find 25. 00-3 inding an Elementary Matrix In Exercises 9-12, t A, B, and C be 27. 2 Fine In and mat 29. 31. . Find an elementary matrix Fsuch that FA-R a. Find an elementary matris Fsuch that FA-r 1. Find an elementary matris Fsuch that FR-A 2 Find an elementary matrix F such that Fr 33. inding a Sequence of Elementary Matrices Exercises 13-18, find a sequence of elementary matrices at can be used to write the matrix in row-echelon form. 35 3 -3 14. -1 -2 37. 10 -5 2 I -2 -1 38. 8 -4 16. 2 3 5-1 12 -2 I 3 -4 I -2 -3 5 -1 4 8-5 18.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Done elementary_linear_algebra_8th_...
82
Chapter 2 Matrices
2.4 Exercises Se Cuichat com tor woted ou solutona to odd numbered enercises.
Elementary Matrices In Exercises 1-8, determine
whether the matrix is elementary. If it is, state the
elementary row operation used to produce it.
Finding the Inverse of an Elementary Matrix
In Exercises 19-24, find the inverse of the elementary
matrix.
1.
2.
lo
20. 5
19.
3.
12
[o
21. 0
4.
ol
[2
5. 0
22. 0
0 -3
0]
6. 0
[2
[A
23. 0 1
0 0
24.
.
0 0
7.
-5
1
0 0 0
0 0
Finding the Inverse of a Matrix In Exercises 25-28,
find the inverse of the matrix using elementary matrices.
1.
8.
3 -2
25.
26.
0 -3
0 -1
27. 0 6
6 -1
Finding an Elementary Matrix In Exercises 9-12,
let A, B, and C be
28. 0 2
4
2
-1
2
B =
0.
Finding a Sequence of Elementary Matrices
In Exercises 29-36, find a sequence of elementary
matrices whose product is the given nonsingular matrix.
2
21. and
-1
20. l
31. [;
30. ol
4
29.
C =
-1
2
32.
9. Find an elementary matrix E such that FA - R
10. Find an elementary matrix E such that FA - C
I1. Find an elementary matrix F such that FR- A
12. Find an elementary matrix F such that FC- A
[-2
31
2
33. -1 3
34. 2
5
6.
3
4
[4
2
Finding a Sequence of Elementary Matrices
In Exercises 13-18, find a sequence of elementary matrices
that can be used to write the matrix in row-echelon form.
0 -1
3
35.
0 0
36. o
0 -1
0 0 -2
2
2
| -1
3 -3
13. 0 1 ?1
10 -
14. I -1
2 -2
37. Writing Is the product of two elementary matrices
always elementary? Explain.
38. Writing E is the elementary matrix obtained by
interchanging two rows in A is an x matrix.
(a) How will FA compare with A? (b) Find E.
2
2
-2 -1
0]
[1
8 -4 16. 2
3
3
15.
4
5 -1
-4
2
-6
12
8
-2
-2
3 -4
I -2
-6
39. Use elementary matrices to find the inverse of
3
17.
-3
3
18.
2
4
[1
A-0
2
5 -1
- 1
2 -2
8 -5
ongm
2.4 Exercises
83
40. Use elementary matrices to find the inverse of
53. Determine a and h such that A is idempotent.
A-0
1||o
54. Guided Proof Prove that A is idempotent if and only
if A is idempotent.
Getting Started: The phrase "if and only if" means that
you have to prove two statements:
1. If A is idempotent, then A' is idempotent.
2. If A' is idempotent, then A is idempotent.
(i) Begin your proof of the first statement by assuming
that A is idempotent.
(ii) This means that A- A.
(iii) Use the properties of the transpose to show that A
is idempotent.
(iv) Begin your proof of the second statement by
assuming that A" is idempotent.
True or False? In Exercises 41 and 42, determine
whether each statement is true or false. If a statement
is true, give a reason or cite an appropriate statement
from the text. Ifa statement is false, provide an example
that shows the statement is not true in all cases or cite an
appropriate statement from the text.
41. (a) The identity matrix is an elementary matrix.
(b) If E is an elementary matrix, then 2E is an
elementary matrix.
(c) The inverse of an elementary matrix is an
elementary matrix.
42. (a) The zero matrix is an elementary matrix.
(b) A square matrix is nonsingular when it can be
written as the product.of elementary matrices
55. Proof Prove that ifA is an nXH matrix that is
idempotent and invertible, then A -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2F8e5d5aa1-b018-4210-aa2e-4f9b5d2a80f0%2Fzwe54x9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2:47
Done elementary_linear_algebra_8th_...
82
Chapter 2 Matrices
2.4 Exercises Se Cuichat com tor woted ou solutona to odd numbered enercises.
Elementary Matrices In Exercises 1-8, determine
whether the matrix is elementary. If it is, state the
elementary row operation used to produce it.
Finding the Inverse of an Elementary Matrix
In Exercises 19-24, find the inverse of the elementary
matrix.
1.
2.
lo
20. 5
19.
3.
12
[o
21. 0
4.
ol
[2
5. 0
22. 0
0 -3
0]
6. 0
[2
[A
23. 0 1
0 0
24.
.
0 0
7.
-5
1
0 0 0
0 0
Finding the Inverse of a Matrix In Exercises 25-28,
find the inverse of the matrix using elementary matrices.
1.
8.
3 -2
25.
26.
0 -3
0 -1
27. 0 6
6 -1
Finding an Elementary Matrix In Exercises 9-12,
let A, B, and C be
28. 0 2
4
2
-1
2
B =
0.
Finding a Sequence of Elementary Matrices
In Exercises 29-36, find a sequence of elementary
matrices whose product is the given nonsingular matrix.
2
21. and
-1
20. l
31. [;
30. ol
4
29.
C =
-1
2
32.
9. Find an elementary matrix E such that FA - R
10. Find an elementary matrix E such that FA - C
I1. Find an elementary matrix F such that FR- A
12. Find an elementary matrix F such that FC- A
[-2
31
2
33. -1 3
34. 2
5
6.
3
4
[4
2
Finding a Sequence of Elementary Matrices
In Exercises 13-18, find a sequence of elementary matrices
that can be used to write the matrix in row-echelon form.
0 -1
3
35.
0 0
36. o
0 -1
0 0 -2
2
2
| -1
3 -3
13. 0 1 ?1
10 -
14. I -1
2 -2
37. Writing Is the product of two elementary matrices
always elementary? Explain.
38. Writing E is the elementary matrix obtained by
interchanging two rows in A is an x matrix.
(a) How will FA compare with A? (b) Find E.
2
2
-2 -1
0]
[1
8 -4 16. 2
3
3
15.
4
5 -1
-4
2
-6
12
8
-2
-2
3 -4
I -2
-6
39. Use elementary matrices to find the inverse of
3
17.
-3
3
18.
2
4
[1
A-0
2
5 -1
- 1
2 -2
8 -5
ongm
2.4 Exercises
83
40. Use elementary matrices to find the inverse of
53. Determine a and h such that A is idempotent.
A-0
1||o
54. Guided Proof Prove that A is idempotent if and only
if A is idempotent.
Getting Started: The phrase "if and only if" means that
you have to prove two statements:
1. If A is idempotent, then A' is idempotent.
2. If A' is idempotent, then A is idempotent.
(i) Begin your proof of the first statement by assuming
that A is idempotent.
(ii) This means that A- A.
(iii) Use the properties of the transpose to show that A
is idempotent.
(iv) Begin your proof of the second statement by
assuming that A" is idempotent.
True or False? In Exercises 41 and 42, determine
whether each statement is true or false. If a statement
is true, give a reason or cite an appropriate statement
from the text. Ifa statement is false, provide an example
that shows the statement is not true in all cases or cite an
appropriate statement from the text.
41. (a) The identity matrix is an elementary matrix.
(b) If E is an elementary matrix, then 2E is an
elementary matrix.
(c) The inverse of an elementary matrix is an
elementary matrix.
42. (a) The zero matrix is an elementary matrix.
(b) A square matrix is nonsingular when it can be
written as the product.of elementary matrices
55. Proof Prove that ifA is an nXH matrix that is
idempotent and invertible, then A -
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