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

Keywords:

determinant; GCD matrix; gcd-closed sets

Zbl 0703.15012