Let $A$ be an invertible $3\times3$ matrix with complex values. Prove that:
$$\det\left(I + A^TA^{-1}\right) = 2\left(1 + \operatorname{tr}\left(A^TA^{-1}\right)\right)$$

I've tried to solve this problem with characteristic polynomial but in the end I got stuck in an expression involvind Newton's identities regarding that polynomial. I cannot relate $A^TA^{-1}$ with characteristic polynomial.