Finding the Eigenvalues and Eigenvectors of $(A+I)^{-1}$. (Without doing the actual inversion)

Question

Given a matrix $$A+I = \left[\matrix{2 &0& 2\\ 0& 0 &-2\\ 2 &-2 &1}\right].$$ How would we find the eigenvalues and eigenvectors of the inverse of $(A+I)$? I want to do this without computing the actual inverse.

Answer 1

The eigenvalues of $B^{-1}$ are the reciprocals of the eigenvalues of $B$.

Proof. Suppose $B^{-1}v=\lambda v$; then $\lambda\ne0$ and $\lambda^{-1}v=Bv$. Conversely, if $Bv=\mu v$, then $\mu\ne0$ and $\mu^{-1}v=B^{-1}v$. QED

If you know the eigenvalues of $A+I$, then you also know those of $(A+I)^{-1}$.

Answer 2

Hint: write down the definition of "$\lambda$ is an eigenvalue of $(A+I)^{-1}$", then multiply by $(A+I)$ on both sides of the equation that you will get. If you did it right, all you need is the eigenvalues of $A$ (or of $A+I$, which are related to those of $A$ by a very simple relation).