Left and right multiplying of matrices I am new to matrix multiplication and trying to understand something. Suppose you have a matrix equation $ A x=b $. I know to solve for $x$ you should left multiply by the inverse of A. But what is the reason you can't solve for $b$ like this: $ A A^{-1} x=b A^{-1} $ so that  $ x=b A^{-1} $? 
I tried with an example to see that it doesn't work but I don't have a good understanding of the mechanics why not.
What if you had something like this instead:
$ A x = By $
could you solve like this?
$ AA^{-1} x = BA^{-1}y $
and then get the solution 
$  x = BA^{-1}y $
or do you have to solve like this
$ A^{-1}A x = A^{-1}By $
to get the solution
$  x = A^{-1}By $

You are asking why $AB=A'B'$ does not imply $AEB = A'EB'$. You noticed that the thing is false. So you should not ask yourself "why is false" (which is nonsense) but you should ask yourself "why I thought it could be true?".
You know that applying the same operation to the same object will held the same result. So if $A=B$ you can right multiply and obtain $AC=BC$ or you can left multiply and obtain $CA=CB$. If multiplication were commutative you could also multiply "in the middle" since the order of factors is not relevant. But when the operation is not commutative this is not admissible. Adding a factor in the middle of a product as in $AB \to AEB$ is not an operation performed on the object $AB$ itself, so there is no reason to think that $AB=A'B'$ would imply $AEB \neq A'EB'$.
