A determinantal version of the frobenius-könig theorem
- 1 December 1984
- journal article
- research article
- Published by Taylor & Francis in Linear and Multilinear Algebra
- Vol. 16 (1) , 155-165
- https://doi.org/10.1080/03081088408817619
Abstract
Let F be a field and let {d 1,…,dk } be a set of independent indeterminates over F. Let A(d 1,…,dk ) be an n × n matrix each of whose entries is an element of F or a sum of an element of F and one of the indeterminates in {d 1,…,dk }. We assume that no d 1 appears twice in A(d 1,…,dk ). We show that if det A(d 1,…,dk ) = 0 then A(d 1,…,dk ) must contain an r × s submatrix B, with entries in F, so that r + s = n + p and rank B ⩽ p − 1: for some positive integer p.Keywords
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