A matrix $D$ is calculated as $A*B*C$.
I need to find the matrix $B$ given matrices $A$, $C$ and $D$.
After some trial and error it seems that the following equation is needed to reproduce matrix $B$ whilst retaining the correct order of transformations:
$B=A^{-1}*D*C^{-1}$
Why does this work?
Remembering that matrix multiplication is not commutative, that if $A=B$ then $AC=BC$ (it is not the case that if $A=B$, then $AC=CB$), and that $A*A^{-1}=I$, we can do the following: $$A*B*C=D\\A*B*C*C^{-1}=D*C^{-1}\\A*B=D*C^{-1}\\A^{-1}*A*B=A^{-1}*D*C^{-1}\\B=A^{-1}*D*C^{-1}$$ Remember that this only works if matrices $A$ and $C$ are invertible.