Test 4 Solutions

Test 4 Solutions - Test 4 Solutions MATH1104 Fall 2009 Last...

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Test 4, Solutions MATH1104, Fall, 2009 Last name: First name: Student no.: ——————————————————————————————- 1-[2+4 marks]: Let A = 0 - 4 - 6 - 1 0 - 3 1 2 5 . (i) Find the characteristic polynomial of A . (ii) Find all eigenvalues of A . solution : (i) det A - λI = ( λ - 1)( λ - 2) 2 = 0. (ii) The eigenvalues are λ = 2 and λ = 1. 2-[8+4 marks]: Let A = 4 - 1 6 2 1 6 2 - 1 8 . You are given that λ 1 = 2 and λ 2 = 9 are the eigenvalues of A . (i) Find a basis of the eigenspace corresponding to each eigenvalue. (ii) Is A diagonalizable? If yes, ﬁnd an invertible matrix P and a diagonal matrix D such that A = PDP - 1 ; if no, explain why. Solution: (i) When λ = 2, A - λI = A - 2 I = 2 - 1 6 2 - 1 6 2 - 1 6 R 3 - R 1 ,R 2 - R 1 ------------→ 2 - 1 6 0 0 0 0 0 0 . From ( A - λI ) ~x = ~ 0 we imply 2 x 1 - x 2 + 6 x 3 = 0 , x 2 = 2 x 1 + 6 x 3 . Hence

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Test 4 Solutions - Test 4 Solutions MATH1104 Fall 2009 Last...

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