7.Suppose that A is an n x n matrix.a.For each eigenvalue λ of A, prove that λ² is an eigenvalue of the matrix A².What is the eigenvector corresponding to X²?b.For each eigenvalue λ of A, prove that X-¹ is an eigenvalue of the matrix A-¹(you can assume A is invertible). What is the eigenvector corresponding to \¯¹?C.For each eigenvalue λ of A, prove that λ + c is an eigenvalue of the matrixA + CI, where c is a constant number and I is the n × n identity matrix. What is the eigenvectorcorresponding to λ + c? Prove that the algebraic multiplicity of the corresponding eigenvalue ofA is the same as the algebraic multiplicity of the eigenvalue λ + c of A+ cI.

Answered: Image /qna-images/answer/8bb48fbc-c8ca-420e-ab33-d6af79b07143.jpg

7. Suppose that A is an n x n matrix. a. For each eigenvalue λ of A, prove that λ² is an eigenvalue of the matrix A². What is the eigenvector corresponding to X²? b. For each eigenvalue λ of A, prove that X-¹ is an eigenvalue of the matrix A-¹ (you can assume A is invertible). What is the eigenvector corresponding to X-¹?

7. Suppose that A is an n x n matrix. a. For each eigenvalue λ of A, prove that λ² is an eigenvalue of the matrix A². What is the eigenvector corresponding to X²? b. For each eigenvalue λ of A, prove that X-¹ is an eigenvalue of the matrix A-¹ (you can assume A is invertible). What is the eigenvector corresponding to X-¹?

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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