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QUESTION 5 Given the matrix A = 1 1 -3 -1 3 1 1 -1. a) Find all the eigenvalues of the matrix A. b) Find the smallest eigenvalues of A5.

QUESTION 5 Given the matrix A = 1 1 -3 -1 3 1 1 -1. a) Find all the eigenvalues of the matrix A. b) Find the smallest eigenvalues of A5.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.1: Introduction To Eigenvalues And Eigenvectors
Problem 35EQ
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Do 5) a&b

 

QUESTION 5 Given the matrix A = -1 (19 3 -3 -1 a) Find all the eigenvalues of the matrix A. b) Find the smallest eigenvalues of A5. c) Find the eigenvalues of A¹. d) Find the eigenvalues of A-¹ which are positive. e) Determine the basis for the eigenspace that corresponds to the largest eigenvalue of the matrix A. f) Given the basis for the eigenspace corresponding to eigenvalues λ = -2 and λ = 2 of the matrix A are 4 and {}} respectively. i) Is the matrix A diagonalizable? Give reason(s) for your answer. ii) If matrix A is diagonalizable, state the matrices P and D such that P-¹AP = D. g) Without using a determinant method, give reason(s) why matrix A is invertible?
Transcribed Image Text:QUESTION 5 Given the matrix A = -1 (19 3 -3 -1 a) Find all the eigenvalues of the matrix A. b) Find the smallest eigenvalues of A5. c) Find the eigenvalues of A¹. d) Find the eigenvalues of A-¹ which are positive. e) Determine the basis for the eigenspace that corresponds to the largest eigenvalue of the matrix A. f) Given the basis for the eigenspace corresponding to eigenvalues λ = -2 and λ = 2 of the matrix A are 4 and {}} respectively. i) Is the matrix A diagonalizable? Give reason(s) for your answer. ii) If matrix A is diagonalizable, state the matrices P and D such that P-¹AP = D. g) Without using a determinant method, give reason(s) why matrix A is invertible?
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