assign10 - A then n is an eigenvalue of A n How are the...

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Math 136 Assignment 10 Not To Be Handed In 1. By checking whether columns of P are eigenvectors of A , determine whether P diagonalizes A . If so, determine P - 1 , and check that P - 1 AP is diagonal. a) A = ± 4 2 - 5 3 ² , P = ± 1 3 - 1 1 ² . b) A = ± 1 3 3 1 ² , P = ± 1 1 1 - 1 ² . 2. Let A and B be similar matrices. Prove that: a) A and B have the same eigenvalues. b) tr A = tr B . c) A n is similar to B n for all positive integers n . 3. For each of the following matrices, determine the eigenvalues and corresponding eigenvectors and hence determine if the matrix is diagonalizable. If it is, write the diagonalizing matrix P and the resulting matrix D . a) A = ± 4 - 1 - 2 5 ² b) B = ± 2 1 - 1 4 ² c) C = ± - 2 2 - 3 5 ² d) E = ± 2 2 - 3 - 5 ² . e) F = 4 2 2 2 4 2 2 2 4 f) G = 3 1 1 - 4 - 2 - 5 2 2 5 g) H = - 4 6 6 - 2 2 4 - 1 3 1 h) J = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i) K = 1 6 3 0 - 2 0 3 6 1 j) M = - 3 2 1 4 - 2 - 4 - 9 2 7 4. Show that if λ is an eigenvalue of
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Unformatted text preview: A , then λ n is an eigenvalue of A n . How are the corresponding eigenvectors related? 1 2 5. Show that if A is invertible and ~v is an eigenvector of A , then ~v is also an eigenvector of A-1 . how are the corresponding eigenvalues related? 6. Find a matrix 2 × 2 matrix A that has eigenvalues 2 and 3 with corresponding eigenvectors ± 1 2 ² , ± 1 3 ² respectively. 7. Is it true that every invertible matrix is diagonalizable? Justify your answer. 8. Let A = ± 4-1-2 5 ² . Calculate A 3 . Verify your answer by computing A 3 directly. 9. Let A = ±-2 2-3 5 ² . Calculate A 200 . 10. Let A = ± 2 2-3-5 ² . Calculate A 100 ....
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This note was uploaded on 05/04/2010 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.

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assign10 - A then n is an eigenvalue of A n How are the...

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