If each eigenvalueof $A$ is either $+1$ or $-1$ $ \Rightarrow$ $A$ is similar to ${A^{ - 1}}$
Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?
Necessarily, $A$ is invertible and the eigenvalues of $A^{-1}$ are also $\pm 1$. For every positive integer $k$, $rank((A+\epsilon I)^k)=rank(A^{-k}(A+\epsilon I)^k)=rank((A^{-1}+\epsilon I)^k)$ where $\epsilon \in\{\pm1\}$; we conclude using the Jordan test of similarity.