$A$ being similar to $B$ means that there exists an $S$ such that $A=\inverse{S}BS$. So, $B=SA\inverse{S}$ and because $S$, $A$, and $\inverse{S}$ are nonsingular, by Theorem NPNT, $B$ is nonsingular.
\begin{align*}
\inverse{A}
&= \inverse{\left(\inverse{S}BS\right)}
&&\knowl{./knowls/definition.SIM.knowl}{\text{Definition SIM}}\\
&= \inverse{S}\inverse{B}\inverse{\left(\inverse{S}\right)}
&&\knowl{./knowls/theorem.SS.knowl}{\text{Theorem SS}}\\
&= \inverse{S}\inverse{B}S
&&\knowl{./knowls/theorem.MIMI.knowl}{\text{Theorem MIMI}}
\end{align*}
Then by Definition SIM, $\inverse{A}$ is similar to $\inverse{B}$.