Answered: Let Aand B…

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 41EQ

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Let Aand B ben×nmatrices,andassumethatvinRn isaneigenvectorof Acorrespondingtotheeigenvalue λ and also an eigenvector of B corresponding to the eigenvalue µ.
Q. Prove that v is an eigenvector of the matrix AB. What is the corresponding eigenvalue?

Expert Solution

Step 1

$A$ and $B$ be $n \times n$ matrices and $v \in ℝ^{n}$ is an eigenvector of $A$ corresponding to the eigenvalue $λ$ and also an eigenvector of $B$ corresponding to the eigenvalue $μ$ .

We have to prove that $v$ is an eigenvector of the matrix $A B$ .

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