201 Quiz 6 TH solutions

Linear Algebra with Applications (3rd Edition)

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110.201 Linear Algebra 6th Quiz May 5, 2005 Problem 1 Let A = ± 1 4 1 - 2 ² . 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 The eigenvalues are 2 and - 3, and we may choose S = ± 4 1 1 - 1 ² , with D = ± 2 0 0 - 3 ² . Next, A n = S - 1 D n S = ± 1 / 5 1 / 5 1 / 5 - 4 / 5 ² ± 2 n 0 0 ( - 3) n ² ± 4 1 1 - 1 ² = ± 4 * 2 n / 5 + ( - 3) n / 5 2 n / 5 - ( - 3) n / 5 4 * 2 n / 5 - 4 * ( - 3) n / 5 2 n / 5 + 4 + ( - 3) n / 5 ² . Problem 2 Let A = ± 0 - i 2 i 1 ² , i 2 = - 1 . 1. Find the eigenvalues of A. 2 Find A 30 . Solution The eigenvalues are 2 and - 1. We may take S = ± - i i 2 1 ² . To compute A 30 , A 30 = S - 1 D 30 S = ± i/ 3 1 / 3 - 2 i/ 3 1 / 3 ² ± 2 30 0 0 1 ² ± - i i 2 1 ² ± 2 30 / 3 + 2 / 3 1 / 3 - 2 30 / 3 2 / 3 - 2 * 2 30 / 3 2 * 2 30 / 3 + 1 / 3 ² . Problem 3 Let A to be a 3 × 3 matrix with eigenvalues 1, - 1, 2. Let B = A 3 - 5 A 2 . 1
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1. Find the eigenvalues of B. Is B diagonalizable? If yes, find a diagonal
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This homework help was uploaded on 01/23/2008 for the course MATH 201 taught by Professor Consani during the Spring '05 term at Johns Hopkins.

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201 Quiz 6 TH solutions - 110.201 Linear Algebra 6th Quiz...

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