A bijective proof of generalized Cauchy–Binet, Laplace, Sylvester and Dodgson formulas

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].

Keywords:

• Determinant of rectangular matrix
• generalized Cauchy–Binet formula
• generalized Laplace's formula
• generalized Sylvester's formula
• generalized Dodgson's formula

2010 Mathematics Subject Classifications:

• 15A15
• 05A15
• 05C50

Acknowledgments

We would like to express our gratitude to Dr J. Rooin and H. Teimoori for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).