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QUESTION 5 Given the matrix A = -1 (10) 3 -3 a) Find all the eigenvalues of the matrix A. b) Find the smallest eigenvalues of A5.

QUESTION 5 Given the matrix A = -1 (10) 3 -3 a) Find all the eigenvalues of the matrix A. b) Find the smallest eigenvalues of A5.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Do 5e only

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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