Matrix Division

Question

Somebody said matrix division isn’t a thing and there’s nothing I hate more than being told that something in maths or physics can’t be invented to meet that specific goal or result. In this case I was told division isn’t a thing but I know that I can exponentiate a matrix. Now e^A is in itself a matrix if A is an n by n matrix. This is likewise for e^B. Now we know how to deal with e^A/e^B this is just e^(A-B) and what we notice is that we just defined the division of matrices very nicely through the concept of exponentiation. My question is, would this work?