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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.