The following proof, which appears in the article *[A Short and Elementary Proof of the Two-sidedness of the Matrix-Inverse][1]* [Paparella, P., *College Math. J.* **48** (2017), no. 5, 366-367; MR3723713], and only uses the criteria stipulated by the OP: 

[![enter image description here][2]][2]


 [1]: http://www.jstor.org/stable/10.4169/college.math.j.48.5.366
 [2]: https://i.sstatic.net/7xOpn.png