So my questions is let's say you were asked to solve for $A$, and you have something like this:
$$BAC=D$$ where B, C , and D are matrices. So the way I would solve this would be to multiply both sides by $B^{-1}$ and $C^{-1}$ (inverse of B and C), but since the order in the multiplication matters $A = DB^{-1}C^{-1}$ would be different than say $A = B^{-1}DC^{-1}$. My question (maybe stupid or I am just missing something) is how do you know which order is the correct one?
Best Answer
You must be sure to multiply on the correct side. To get rid of the $B$ in $BAC$, you must multiply on the left by $B^{-1}$, so you must do the same on the righthand side of the equation:
$$AC=B^{-1}BAC=B^{-1}D\;.$$
To get rid of the $C$ in $AC$, you must multiply $AC$ on the right by $C^{-1}$, so you must do the same thing on the other side of the equation:
$$A=ACC^{-1}=B^{-1}DC^{-1}\;.$$