Trace of the inverse of a matrix

Question

I am wondering if there is any quick way to find the trace of the inverse of a matrix A = $\operatorname{tr}(A^{-1})$ without using eigenvalues. I just read that $\operatorname{Tr}(A^{-1})\geq n^2\operatorname{Tr}(A)^{-1}$ in a post, so it can be useful to have an idea of $\operatorname{tr}(A^{-1})$ in a MCQ for example. But is there a trick to have to right value without finding the whole inverse (it becomes tough to do it with large matrices) ?