2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/125310Sylvester'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.28 pages, 8 figuresCombinatorics05A30, 15A15text