given the inverse of a matrix, is there an efficient way to find the determinant?

Solution 1:

This is strictly harder than a problem we already don't know how to solve: quickly taking the determinant of an orthogonal matrix.

If $Q$ is an orthogonal matrix, then $Q Q^{\mathsf T} = I$. In particular, if you're given $Q$, you are also automatically given its inverse. The determinant $\det(Q)$ is guaranteed to be $\pm 1$; however, determining which one it is appears to be just as hard as taking any other determinant.

If we could, in general, efficiently compute $\det(A)$ given both $A$ and $A^{-1}$, then in particular, for an orthogonal matrix $Q$, we could efficiently compute $\det(Q)$.