1 0, 1 5. 0 0 1 010

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Please solve parts 45-48 of this problem... Take as much time as you need... linear algebra review

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
0
2 0
4(:D) 2 (1:5) + (1) 4(:)
41.
1 00
42.
100
43. 200
100
0 10
020
010
0 1
45. 001
01 0
53.
57.
46.
0 1 1
101
1 10
04
001
010
0
01
020
49.
.. (1) 50 (1) 44(:¦:-) (9)
0
03
51. 20-1
52.
401
20
0
01 0
0 1
0
54.
-1
47. 0 02
0 20
32 2
232
223
55.
48.
1 1 0
0 1
0 1 1
01 0
004
0 1 0
56.
1 1
1 -2 1
-2 1
01 0
0 1 0
01 0
0 50
(D)(-+) (9) - (-)
1 1 1
58. 1 -1 1
59. 5 1 1
60. 1 -1 1
0 1 0
0 10
01 0
0 10
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 0 2 0 4(:D) 2 (1:5) + (1) 4(:) 41. 1 00 42. 100 43. 200 100 0 10 020 010 0 1 45. 001 01 0 53. 57. 46. 0 1 1 101 1 10 04 001 010 0 01 020 49. .. (1) 50 (1) 44(:¦:-) (9) 0 03 51. 20-1 52. 401 20 0 01 0 0 1 0 54. -1 47. 0 02 0 20 32 2 232 223 55. 48. 1 1 0 0 1 0 1 1 01 0 004 0 1 0 56. 1 1 1 -2 1 -2 1 01 0 0 1 0 01 0 0 50 (D)(-+) (9) - (-) 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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