The ring of matrix is not an integral domain. How to prove that the inverse is unique?
If $ab=ba=1$, and if also $ac=ca=1$, you have $c=(ba)c=b(ac)=b$. In fact, if you reread it, this shows that whenever you know both a left inverse exists and a right inverse exists, then actually they are the same element, so it is a two-sided inverse and it is unique.
It has little to do with integral domains: it's true for any associative operation with an identity.