Let $A$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ 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 ${A^{ - 1}}$?
Characteristic equation of the matrix $A$ is of the form $(x-1)^a(x+1)^b$.
Since it is product of linear factors, it is similar to a matrix in Jordan form. You can continue from here
The eigenvalues of $A^{-1}$ are the reciprocals of the eigenvalues of $A$. By definition, similair matrices have the same eigenvalues.  

The proof that $A^{-1}$ and $A$ have the same eigenvalues is simple:
$$Ax=\lambda x \iff A^{-1}Ax = A^{-1} \lambda x \iff \frac{1}{\lambda} x = A^{-1}x$$