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.