How to prove that the inverse of a matrix is unique?

The ring of matrix is not an integral domain. How to prove that the inverse is unique?

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You don't need it to be an integral domain to have unique inverses. – mez May 22 '14 at 18:34

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.