Can Sylvester's determinantal identity, equivalently Muir's law of extensible minors be generalized?

Abstract

We show that Sylvester's classical determinantal identity is equivalent to its generalizations given by the Mühlbach–Gasca–(López-Carmona)–Ramírez identity and the Beckermann–Mühlbach identity. We also show that the generating extension principles associated with each of these three identities (Muir's law of extensible minors, Mühlbach's extension principle and the extension principle given in Camargo, 2013 [3]) are equivalent and we use this fact to conclude that all these six statements (and also Jacobi's identity on the minors of the adjugate and Cayley's law of complementaries) are actually equivalent. These findings lead naturally to enquire whether there would exist a generalization of Sylvester's identity (or an extension principle) which could not be derived from Sylvester's identity itself.