On some determinantal identities formation laws.

We show that Muir’s law of extensible minors, Cayley’s law of complementaries and Jacobi’s identity for minors of the adjugate [Determinantal identitiesLinear Algebra and its Applications52/53 (1983) pp. 769–791] are equivalent. We also show our generalization of Mühlbach/Muir’s extension principle [A generalization of Mühlbach’s extension principle for determinantal identities.Linear and Multilinear Algebra61 (10) (2013) pp. 1363–1376] is equivalent to its previous form derived by Mühlbach. As a corollary, we show that Mühlbach–Gasca–(Lopez-Carmona)–Ramirez identity [A generalization of Sylvester’s identity on determinants and some applications.Linear Algebra and its Applications66 (1985) pp. 221–234/On extending determinantal identities.Linear Algebra and its Applications132 (1990) pp. 145–162] is equivalent to its generalization found by Beckermann and Mühlbach [A general determinantal identity of Sylvester type and some applications.Linear Algebra and its Applications197,198 (1994) pp. 93–112]. [ABSTRACT FROM PUBLISHER]