Test 4 Solutions

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online