Eigenvalues and Eigenvectors Consisting of A and the Inverse of A For the following, assume that there is a basis consisting of eigenvectors of $A$.
Under this assumption, 
How do the eigenvalues of $A$ compare with the eigenvalues of $A^2$?
If $A$ is also an invertible matrix, how do the eigenvalues of $A$ compare with the eigenvalues of $A^{−1}$?

\begin{align}
Ax&=\lambda x\\
AAx&=\lambda Ax\\ 
A^2x&=\lambda (\lambda x)\\
A^2x&=\lambda^2 x\\
\end{align}
Assuming $A^{-1}$ exists,
\begin{align}
Ax&=\lambda x\\
A^{-1}Ax&=\lambda A^{-1}x\\
Ix&=\lambda A^{-1}x\\
A^{-1}x&=\dfrac{1}{\lambda} Ix\\
A^{-1}x&=\dfrac{1}{\lambda} x\\
\end{align}
