Assume that we have matrices $A, B, C, X$ such that both $X$ and $B$ are symmetric and the dimension of all those matrices agree for multiplications. I need to find the derivative of
$\frac{log\,|B^{-1}|}{\partial x_{ij}}$, such that
$x_{ij}$ indicates the elements of $X$, and $A$ and $B$ are constant of $X$
$B=\big(X \otimes A +C\big)^{-1}$
Based on my reading I think the answer is
$\frac{log\,|B^{-1}|}{\partial x_{ij}}= \text{tr}\big((B^{-1})^{-1} \frac{\partial\,B^{-1}}{\partial x_{ij}}\big),$
$\qquad\quad\,\,=\text{tr}\Big[B\,\, \frac{\partial\big(X\otimes A\,+C\big)}{\partial x_{ij}}\Big]$
$\qquad\quad\,\,=\text{tr}\Big[B\,\, \big(\frac{\partial\,X}{\partial x_{ij}}\otimes A\big)\Big]$
It is known that $\frac{\partial\,X}{\partial x_{ij}}=(2-\delta_{ij})\,E_{ij}$, where for a symmetric matrix $\delta_{ij}$ is one for $i=j$ and zero otherwise, while $E_{ij}$ is the matrix with 1 on the $ij^{th}$ position and zero elsewhere.
My question is how can I expand $\Big[B\big(\frac{\partial\,X}{\partial x_{ij}}\otimes A\big)\Big]$, so which matrix should $E_{ij}$ takes its $ij^{th}$ element.