What is an inverse matrix's eigenvalue?

A matrix A has an eigenvalue ${\displaystyle \lambda }$ only if ${\displaystyle A^{-1}}$ has eigenvalue ${\displaystyle {\lambda }^{-1}}$. For example:
$Av=\lambda v\Rightarrow A^{-1}Av=\lambda A^{-1}v\Rightarrow A^{-1}v=\frac{1}{\lambda }v$
If matrix A has eigenvalue ${\displaystyle \lambda }$, then I-A has eigenvalue ${\displaystyle 1-\lambda }$. Therefore, ${\displaystyle {(I-A)}^{-1}}$ has eigenvalue ${\displaystyle \frac{1}{1-\lambda }}$

If you are looking at a single eigenvector v only, with eigenvalue ${\displaystyle \lambda }$, then A just acts as the scalar ${\displaystyle \lambda }$, and any reasonable expression in A acts on v as in ${\displaystyle \lambda }$. This works for expressions I−A, which is really 1−A, so it acts as ${\displaystyle 1-\lambda }$, its inverse (I−A)−1, in fact for any rational function of A (if well defined; this is where you need ${\displaystyle \lambda 1<1}$).