I've been trying to find material on matrix calculus but it seems hard to find ones with understandable proofs.
I'm doing research work and I am trying to verify some computation. Suppose that I have a matrix $A= \left( \begin{array}{ccc} \beta_{11} + c\beta_{12} +\beta_{13} & -c\beta_{12} & -\beta_{13} \\ -c\beta_{12} & c\beta_{12}+\beta_{22}+\beta_{23} & -\beta_{23} \\ -\beta_{13} & \beta_{23} & -\beta_{13}+\beta_{23}+\beta_{33} \end{array} \right) $ where $c$ is a constant, how do I evaluate $\frac{d}{d\beta_{12}}\det{(A)}$?
From what I have searched, if $A= \left( \begin{array}{ccc} \beta_{11} & \beta_{12} & \beta_{13} \\ \beta_{21} & \beta_{22} & \beta_{23} \\ \beta_{31} & \beta_{32} & \beta_{33} \end{array} \right) $ , i.e. no 2 elements are identical, then $\frac{d}{d\beta_{12}}\det{(A)}=\det{(A)}\cdot A^{-1}_{12}$.
What about the former case? Is there some sort of product rule like in 1-variable calculus?
I forgot to mention, the above is just a simplified case of the problem I'm working on. For my case, the matrix $A$ has dimension 300x300
Thanks!