Problem 1. Consider the matrices [1 1 0] A = |0 2 2 0 0 3 1 3 2 A? = |0 4 10 0 0 9 1/3 1/2 -1/3 1/3 1 -1/2 . (a) Find the eigenvalues of A, A² and A-1. (b) Find the eigenspaces of A. (c) Show that the eigenspaces of A are also eigenspaces of A² and A-' (even though the eigenvalues are different).
Problem 1. Consider the matrices [1 1 0] A = |0 2 2 0 0 3 1 3 2 A? = |0 4 10 0 0 9 1/3 1/2 -1/3 1/3 1 -1/2 . (a) Find the eigenvalues of A, A² and A-1. (b) Find the eigenspaces of A. (c) Show that the eigenspaces of A are also eigenspaces of A² and A-' (even though the eigenvalues are different).
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 38EQ
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Linear Methods
![Problem 1. Consider the matrices
[1 1 0]
A = |0 2 2
0 0 3
[1 3
A? = |0 4 10
0 0
1/3]
1 -1/2
1/2 -1/3
1/3
2
A-
9.
(a) Find the eigenvalues of A, A² and A-.
(b) Find the eigenspaces of A.
(c) Show that the eigenspaces of A are also eigenspaces of A² and A-1 (even though
the eigenvalues are different).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b2fa2bf-9757-4381-9fdb-859cf9db9746%2F5f9605c8-a771-4b78-87fc-02b8775616f8%2Fxiaj6e_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1. Consider the matrices
[1 1 0]
A = |0 2 2
0 0 3
[1 3
A? = |0 4 10
0 0
1/3]
1 -1/2
1/2 -1/3
1/3
2
A-
9.
(a) Find the eigenvalues of A, A² and A-.
(b) Find the eigenspaces of A.
(c) Show that the eigenspaces of A are also eigenspaces of A² and A-1 (even though
the eigenvalues are different).
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