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.

What do you mean "determine" the order? The order of multiplication is the order that you want.
If you have $A = B$ and left multiply by $C$ you get: $CA = CB$
If you right multiply it by $C$ you get: $AC = BC$.
It so happens with $I$ that for any $A$ it is: $AI = IA = A$; that is what maybe confuses you in your example.

$$A(X + B)C = I$$
$$A^{-1}A(X + B)C = A^{-1}I$$
$$A^{-1}A(X + B)CC^{-1} = A^{-1}IC^{-1}$$
$$I (X + B) I = A^{-1}IC^{-1}$$
$$X + B = A^{-1}C^{-1}$$

"Reversing" the order doesn't work:
$$A(X + B)C = I$$
$$C^{-1}A(X + B)C = C^{-1}I$$
$$C^{-1}A(X + B)CA^{-1} = C^{-1}IA^{-1}$$
so you see that $A^{-1}C^{-1} = X + B$ but $C^{-1}A^{-1} = C^{-1}A(X + B)CA^{-1}$ which is not the same at all.
