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

Beslin, Scott; Ligh, Steve

Fibonacci Q. 30, No. 2, 157-160 (1992).

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

Citations:

Zbl 0703.15012