Problem III Let A denote the matrix given below. (i) Find all eigenvalues of the matrix A. (ii) For each eigenvalue of A, find an associated eigenvector. (iii) Find a diagonal matrix D and an invertible matrix U such that A = UDU- 41. 1 0 () 100 010 42. 45. 0 1 0 040 () () 001 001 010 0 1 0 1 0 (D) 100 020 46. 0 2 0 43. 200 010 040 -(9) 44. 100 0 1 0 0 -1 10 () () 48. 004 0 20 010 47. 0 02
Problem III Let A denote the matrix given below. (i) Find all eigenvalues of the matrix A. (ii) For each eigenvalue of A, find an associated eigenvector. (iii) Find a diagonal matrix D and an invertible matrix U such that A = UDU- 41. 1 0 () 100 010 42. 45. 0 1 0 040 () () 001 001 010 0 1 0 1 0 (D) 100 020 46. 0 2 0 43. 200 010 040 -(9) 44. 100 0 1 0 0 -1 10 () () 48. 004 0 20 010 47. 0 02
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.1: Introduction To Eigenvalues And Eigenvectors
Problem 38EQ
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![Problem III Let A denote the matrix given below.
(i) Find all eigenvalues of the matrix A.
(ii) For each eigenvalue of A, find an associated eigenvector.
(iii) Find a diagonal matrix D and an invertible matrix U such that A = UDU-¹.
0 1 0
1 0
2 0
4
41.
-(:D) - (:D) * () +(:)
1 00
42.
100
43. 200
100
0 10
020
0 1
45. 0 0 1
01 0
53.
57.
46.
0 1 1
10 1
1 10
04
001
010
0
01
020
49.
.. (1) 50 (1) 4(:¦:-) (9)
0
103
51. 2 0 -1
52.
401
2
01 0
010,
-1
47. 0 02
20
0
32 2
54. 232
223
55.
48.
110
0 1
0 1 1
010
004
0 1 0
56.
1
1
1 -2 1
-2 1
01 0
0 1 0
01 0
0 50
(D)(-+) (9) -(-5)
1 1 1
58. 1 -1 1
59. 5 1 1
60. 1
-1 1
0 1 0
0 10
01 0
0
10](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47d370c3-9e4b-442d-9a89-d591c5ced338%2Fe2e7b52b-dbec-47d3-86e3-0928a148ce08%2Fy8lsn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem III Let A denote the matrix given below.
(i) Find all eigenvalues of the matrix A.
(ii) For each eigenvalue of A, find an associated eigenvector.
(iii) Find a diagonal matrix D and an invertible matrix U such that A = UDU-¹.
0 1 0
1 0
2 0
4
41.
-(:D) - (:D) * () +(:)
1 00
42.
100
43. 200
100
0 10
020
0 1
45. 0 0 1
01 0
53.
57.
46.
0 1 1
10 1
1 10
04
001
010
0
01
020
49.
.. (1) 50 (1) 4(:¦:-) (9)
0
103
51. 2 0 -1
52.
401
2
01 0
010,
-1
47. 0 02
20
0
32 2
54. 232
223
55.
48.
110
0 1
0 1 1
010
004
0 1 0
56.
1
1
1 -2 1
-2 1
01 0
0 1 0
01 0
0 50
(D)(-+) (9) -(-5)
1 1 1
58. 1 -1 1
59. 5 1 1
60. 1
-1 1
0 1 0
0 10
01 0
0
10
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