I am not sure where to begin with this question but any guidance will be greatly appreciated:
Question: Let A be a nonsingular matrix and let lambda be an eigenvalue of A. Show that 1/lambda is an eigenvalue of A^-1
If $\displaystyle A\in \mathbb{K}^{n\times n}$ is nonsingular and $\displaystyle \lambda \in \mathbb{K}$ is an eigenvalue of $\displaystyle A$ then, $\displaystyle \lambda\neq 0$ (why?) and there exists $\displaystyle 0\neq x \in \mathbb{K}^n$ column vector such that $\displaystyle Ax=\lambda x$ . Now, multiply both sides by $\displaystyle A^{-1}$ .
Fernando Revilla