Why there is no operation of matrix division?

Question

I was just thinking that why we dont divide matrices by matrices, i understand that this is weird but i dont have a clear answer yet .

Is there any intuitive answer ? just like we cant have a dot product of a vector with a scalar .

Any kind of help is great.

Answer 1

There is, but only in certain cases. As with real numbers, just define $$A\div B \equiv AB^{-1}$$ whenever this expression makes sense (i.e., when $B^{-1}$ exists and is compatible as a right multiplier of $A$).

Answer 2

1. Not all matrices are invertible.
2. For invertible matrices, multiplication between them do not commute in general. So 'division' does not commute as well. Thus, we avoid the notion of division in a non-commutative ring.