Let $\displaystyle \lambda$ be an eigenvalue of a square matrix $\displaystyle A$.

How to show that:

1. $\displaystyle \lambda^n$ is an eigenvalue of $\displaystyle A^n$ where n is a positive integer.

2. If $\displaystyle A$ is invertible, show that $\displaystyle \frac {1}{\lambda}$ is an eigenvalue of $\displaystyle A^{-1}$.

To start off: I was told to let $\displaystyle x$ be an eigenvector such that $\displaystyle Ax= \lambda x$

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