Diagonalization & Matrix Powers · Superchallenge problem

A real $3\times 3$ matrix $A$ satisfies $\operatorname{tr}A = 5$, $\operatorname{tr}(A^2) = 21$ and $\operatorname{tr}(A^3) = 71$, and $A$ is invertible. Find $\operatorname{tr}(A^{-1})$.