Let $\displaystyle A \in M_n(\mathbb{R} )$ and let $\displaystyle T(A) = \left( A^{-1} \right) ^T $.

Prove that $\displaystyle trace[\mathcal{D} T(A)] = -trace[A^{-1} \cdot (A^T)^{-1}]$,

Where $\displaystyle \mathcal{D}f(p)$ is the Jacobian of $\displaystyle f(p)$.

Already proved prior to this that the determinant transformation is continuous and that $\displaystyle GL_n( \mathbb{R})$ is an open set. Don't really have a clue how to approach this -- a direction would be nice.

Thanks.