201 Quiz 6 FR solutions

Linear Algebra with Applications (3rd Edition)

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110.201 Linear Algebra 6th Quiz May 6, 2005 Problem 1 Let A = 2 0 1 3 . 1. Find an invertible matrix S and a diagonal matrix D such that S - 1 AS = D . 2. Find a formula for the entries of A n , where n is a positive integer. Solution λ 1 = 2 and λ 2 = 3. S = 1 0 - 1 1 , D = 2 0 0 3 and A = S - 1 DS . Next, A n = S - 1 D n S = 1 0 1 1 2 n 0 0 3 n 1 0 - 1 1 = 2 n 0 2 n - 3 n 3 n . Problem 2 Let A = 1 2 i - i 0 , i 2 = - 1 . 1. Find the eigenvalues of A. 2 Find A 50 . Solution The eigenvalues are 2 and - 1. One choice for S is S = 2 i - i 1 1 . Then A 50 = S - 1 D 50 S = - i/ 3 1 / 3 i/ 3 2 / 3 2 50 0 0 ( - 1) 2 i - i 1 1 = 2 51 / 3 - 1 / 3 - 2 50 / 3 - 1 / 3 - 2 51 - 2 / 3 2 50 / 3 - 2 / 3 . 1

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Problem 3 Let A to be a 3 × 3 matrix with eigenvalues 2, - 2, 1. Let B = A 3 - 3 A 2 . 1. Find the eigenvalues of B. Is B diagonalizable? If yes, find a diagonal matrix D such that D = S
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