Matrix multiplication

Question

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 all three 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

Answer 1

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.

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

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"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.

Answer 2

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.