Let $\displaystyle \lambda$ be an eigenvalue of the nonsingular matrix A with associated eigenvector $\displaystyle \vec x$. Show that $\displaystyle \frac{1}{\lambda}$ is an eigenvalue of $\displaystyle A^{-1}$ with associated eigenvector $\displaystyle \vec x$.

Things I thought about: roots of $\displaystyle det(\lambda I_n-A)$ are eigenvalues, inverse is calculated by rref[A|In]