Jacobian of inverse of matrix $A(x) \in \mathcal{M}_{7\times7}$?

Question

I have a matrix $A(x)$ where $x\in \mathbf{R}^{7}$. I have to calculate $\frac{\partial}{\partial x}A(x)^{-1}$ and then I will evaluate it at some $x_{0}$. Now this matrix is very dense so its not possible to use "Matrix Inversion Lemma" since it requires inverse of submatrices which is not possible.

Any pointers on how to solve this problem?

Thanks

Answer 1

No, the derivative of the inverse of a matrix is given by

$$ \frac{\partial A^{-1}}{\partial x_i} = -A^{-1} \frac{\partial A}{\partial x_i} A^{-1}. $$