Find characteristic polynomial of $T^{-1}$

Question

Given that $[T]_B=\begin{bmatrix}1&0\\0&2\end{bmatrix}$ for some linear transformation $T:V\rightarrow V$, and $B$ basis for $V$, I'm trying to find the characteristic polynomial of $T^{-1}$.

I know that $p_{T^{-1}}(x)=|xI-[T^{-1}]_B|$. Is there a formula for $[T^{-1}]_B$ that uses $[T]_B$?

Answer 1

Yes, there is: $[T^{-1}]_B=[T]_B^{\,-1}$. So, compute the characteristic polynomial of $$\begin{bmatrix}1&0\\0&\frac12\end{bmatrix}.$$