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This is kind of trivial and useless and well-known, but I just realised that if $p(x)$ is the charpoly of matrix/operator $T$ with nonzero constant term $c$, then $q(x):=1-p(x)/c$ has zero constant term, by Cayley-Hamilton $q(T)=1$ and since $q(x)=xr(x)$, we have that $r(T)=T^{-1}$.

So if you know the charpoly of an operator/matrix (hah! as if!) you have a polynomial formula for its inverse.