Sylvester's identity is a classical determinantal identity with a simple linear algebra proof. We present combinatorial proofs of several non-commutative extensions, and find a $\beta$-extension that is both a generalization of Sylvester's identity and the $\beta$-extension of the quantum MacMahon master theorem.
['Matjaz Konvalinka']
Non-commutative Sylvester's determinantal identity