# Linear Algebra with Applications (3rd Edition)

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Chapter 7 / Exercise 170
Precalculus: Real Mathematics, Real People
Larson
Expert Verified
110.201 Linear Algebra 6th Quiz May 5, 2005 Problem 1 Let A = 1 4 1 - 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 ofAn, wherenis a positive integer. 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 . To compute A 30 , A 30 = S - 1 D 30 S = i/ 3 1 / 3 - 2 i/ 3 1 / 3 2 30 0 0 1 - i i 2 1 2 30 / 3 + 2 / 3 1 / 3 - 2 30 / 3 2 / 3 - 2 * 2 30 / 3 2 * 2 30 / 3 + 1 / 3 . 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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Chapter 7 / Exercise 170
Precalculus: Real Mathematics, Real People
Larson
Expert Verified
1. Find the eigenvalues of B. Is B diagonalizable?