## Common Eigenvector of Two Matrices $A, B$ is Eigenvector of $A+B$ and $AB$.

## Problem 382

Let $\lambda$ be an eigenvalue of $n\times n$ matrices $A$ and $B$ corresponding to the same eigenvector $\mathbf{x}$.

**(a)** Show that $2\lambda$ is an eigenvalue of $A+B$ corresponding to $\mathbf{x}$.

**(b)** Show that $\lambda^2$ is an eigenvalue of $AB$ corresponding to $\mathbf{x}$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

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