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# assign10 - Math 136 Assignment 10 Not To Be Handed In 1 By...

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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 , P = 1 3 - 1 1 . b) A = 1 3 3 1 , P = 1 1 1 - 1 . 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 b) B = 2 1 - 1 4 c) C = - 2 2 - 3 5 d) E = 2 2 - 3 - 5 . 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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