Calculating the derivative $\frac{{\partial {{\bf{X}}^{{\rm{ - }}1}}}}{{\partial {\bf{X}}}}$ How to calculate the derivation $\frac{{\partial {{\bf{X}}^{{\rm{ - }}1}}}}{{\partial {\bf{X}}}}$,where ${\bf{X}}$ is square matrix.Thanks a lot for your help! 

First note that $\frac{\partial F}{\partial X}$ is a 4th order tensor. 
Let $F=X^{-1},\,$ then working out the problem in index notation yields
$$\eqalign{
 dF_{ij} &= -F_{ik}\,dX_{kl}\,F_{lj} \cr\cr
\frac{\partial F_{ij}}{\partial X_{kl}} &= -F_{ik}\,F_{lj} \cr\cr
}$$
