Order of Multiplication in Matrix Multiplication So I have a general question on matrix multiplication. I know the order of multiplication matters, except if they're invertible. So, consider something like $A(X+B)C = I$. 
If $\mathbf{A, B, C}$ are invertible, is: $A^{-1}A(X+B)C = A^{-1}$ equivalent to $A(X+B)CA^{-1} = A^{-1}$ ? 
If so, can I do a similar thing for C and say: $X = C^{-1}A^{-1}-B$ ?
Furthermore, that's not necessarily equivalent to $X = A^{-1}C^{-1}-B$ right? 
In general, how do you determine the order of multiplication when simplifying a system? 
Thanks.

In general, don't assume that matrices commute. Almost always, they don't.
Ever if the three matrices $A, B, C$ did have some sort of condition that implied that they commute (perhaps being diagonal), you don't know about $X$ and still can't commute over the $(X + B)$ term. Here, only $X = A^{-1}C^{-1}-B$ is correct.
But to answer your question, "How do you determine the order of multiplication?" I remind you that matrix multiplication is associative, so you can do the 'order' in any way you want as long as you always remember what's on the left and what's on the right. That is, $A(BC) = (AB)C$ and so on.
