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GCD-closed sets and the determinants of GCD matrices. (English) Zbl 0752.11012

For a set \(S=\{x_ 1,x_ 2,\dots,x_ n\}\) of positive integers, the \(n\times n\) matrix \([S]=(s_{ij})\), where \(s_{ij}=(x_ i,x_ j)\), is called GCD matrix on \(S\). In this paper the author obtains a formula for the determinant of a GCD matrix based on a class of gcd-closed sets. Using this formula closed-form expressions for the determinants of some special GCD matrices are obtained. Moreover the formula obtained by Zhongshan Li [Linear Algebra Appl. 134, 137-143 (1990; Zbl 0703.15012)] comes out as a corollary of the above mentioned formula.
Reviewer: S.Lal (Chandigarh)

MSC:

11C20 Matrices, determinants in number theory
15B36 Matrices of integers
15A15 Determinants, permanents, traces, other special matrix functions

Citations:

Zbl 0703.15012
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