Eigenvalues for T if and only if it is also eigenvalue of T inverse

Question

Let $V$ be a finite-dimensional vector space over $\mathbb{F}$ with $T \in \mathcal{L}(V, V)$ invertible and $\lambda \in \mathbb{F} \setminus \{0\}$. Prove that $\lambda$ is an eigenvalue for $T$ if and only if $\lambda^{-1}$ (inverse) is an eigenvalue for $T^{-1}$ (inverse).

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Answer 1

$$Tv = \lambda v \iff v = T^{-1}\lambda v = \lambda T^{-1} v \iff T^{-1}v = \lambda^{-1}v$$