1. Find a matrix A with 25 as an eigenvalue with eigenvector v1 and 0 as an eigenvalue with eigenvector v2 Is your matrix invertible? Is it orthogonally diagonalisable?

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 24EQ
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1. Find a matrix A with 25 as an eigenvalue with eigenvector v1
and 0 as an eigenvalue
with eigenvector V2:
Is your matrix invertible? Is it orthogonally diagonalisable?
2. Let A be a 3 x 3 matrix. Assume 1 and 2 are the only eigenvalues of A.
Determine whether the following statements are always true. If true, justify why. If not
true, provide a counterexample.
Statement A: If vị is an eigenvector of A corresponding to 1 and v2 is an eigenvector
corresponding to 2, then A(vı + v2) = 3(v1 + V2)
Statement B: One of the eigenspaces of A is two-dimensional, and the other is one-
dimensional.
Transcribed Image Text:- 1. Find a matrix A with 25 as an eigenvalue with eigenvector v1 and 0 as an eigenvalue with eigenvector V2: Is your matrix invertible? Is it orthogonally diagonalisable? 2. Let A be a 3 x 3 matrix. Assume 1 and 2 are the only eigenvalues of A. Determine whether the following statements are always true. If true, justify why. If not true, provide a counterexample. Statement A: If vị is an eigenvector of A corresponding to 1 and v2 is an eigenvector corresponding to 2, then A(vı + v2) = 3(v1 + V2) Statement B: One of the eigenspaces of A is two-dimensional, and the other is one- dimensional.
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