Non-commutative Sylvester's determinantal identity

英文摘要：

Sylvester's identity is a classical determinantal identity with a straightforward linear algebra proof. We present a new, combinatorial proof of the identity, prove several non-commutative versions, and find a $β$-extension that is both a generalization of Sylvester's identity and the $β$-extension of the MacMahon master theorem.