E (a) (b) (c) Consider the following matrices -3 4 07 A-2 30 -8 8 1 and B and vector b, H 0 3 1 0 1] B 0 -1 0 0 0 1 Find the characteristic polynomial of matrix A. " Find eigenvalues of the matrix A. b= Find a basis for the eigenspaces of matrix A.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section: Chapter Questions
Problem 9RQ
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Question
►
(a)
(b)
(c)
(f)
(g)
Consider the following matrices
-3 4
0
A-2 30
-88 1
(i)
0
17
0 -1
191.
0
0 0
Find the characteristic polynomial of matrix A.
"
and B and vector b,
G
0
3
B =
Find eigenvalues of the matrix A.
(d)
What is the algebraic and geometric multiplicities of the eigenvalues of A.
(e)
Show that the matrix A is diagonalizable and find an invertible matrix P and
a diagonal matrix D such that P-¹AP = D
b=
Find a basis for the eigenspaces of matrix A.
Find A¹0b by writing b as linear combination of eigenvectors of A.
Find a formula for Ak for all non-negative integers k.
Use (g) to find A¹0b and compare it with what you found in (VI).
Is A similar to B? If yes, find an invertible matrix such that P-¹AP = B.
Transcribed Image Text:► (a) (b) (c) (f) (g) Consider the following matrices -3 4 0 A-2 30 -88 1 (i) 0 17 0 -1 191. 0 0 0 Find the characteristic polynomial of matrix A. " and B and vector b, G 0 3 B = Find eigenvalues of the matrix A. (d) What is the algebraic and geometric multiplicities of the eigenvalues of A. (e) Show that the matrix A is diagonalizable and find an invertible matrix P and a diagonal matrix D such that P-¹AP = D b= Find a basis for the eigenspaces of matrix A. Find A¹0b by writing b as linear combination of eigenvectors of A. Find a formula for Ak for all non-negative integers k. Use (g) to find A¹0b and compare it with what you found in (VI). Is A similar to B? If yes, find an invertible matrix such that P-¹AP = B.
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