Given the characteristic polynomial $\chi\_A$ of an invertible matrix $A$, I'm to find $\chi\_{A^{-1}}$.

I can see that this is theoretically possible. $\chi\_A$ uniquely determines the similarity class of $A$, which uniquely determines the similarity class of $A^{-1}$, which uniquely determines $\chi\_{A^{-1}}$.

Calculating the coefficients of $\chi\_{A^{-1}}$ explicitly and then relating them to the coefficients of $\chi\_A$ seems unfeasible. I thought about calculating the factors instead, which could be easier since I at least have some idea what the linear factors $\chi\_{A^{-1}}$ are (since I can see how to get the eigenvalues of $A^{-1}$ from those of $A$), but that doesn't help me with potential higher-degree irreducibles or repeated linear factors. Also, **we're not supposed to know the eigenvalues of $A^{-1}$**, at least I don't think so, since calculating them is the next question.

Any hints?