ABSTRACT

In this paper, we give the generalization of Cauchy–Binet, Laplace, Sylvester and generalized Dodgson's condensation formulas for the case of rectangular determinants. The proofs are bijective combinatorial proofs similar to that of Zeilberger's paper [Zeilberger D. Dodgson's determinant-evaluation rule proved by TWO-TIMING MEN and WOMEN. Electron J Comb. 1997;4(2):R22; Zeilberger D. A combinatorial approach to matrix algebra. Discrete Math. 1985;56:61–72].

中文翻译：

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广义 Cauchy–Binet、Laplace、Sylvester 和 Dodgson 公式的双射证明

摘要

在本文中，我们给出了 Cauchy-Binet、Laplace、Sylvester 和广义 Dodgson 凝聚公式在矩形行列式的情况下的推广。证明是双射组合证明，类似于 Zeilberger 的论文 [Zeilberger D. Dodgson 的行列式评估规则，由 TWO-TIMING MEN 和 WOMEN 证明。电子 J 梳。1997;4(2):R22; Zeilberger D. 矩阵代数的组合方法。离散数学。1985；56:61-72]。