On the Singularity of LCM Matrices

Abstract

Let S=x1,x2,...,xn be a set of distinct positive integers. The set S is called gcd-closed if it contains the greatest common divisor (xi,xj) of xi and xj for $1\le i,j\le n.$ The matrix [S] is called the least common multiple (LCM) matrix on S if its i,j entry is the least common multiple [xi,xj] of xi and xj. Bourque and Ligh conjectured that the LCM matrix on a gcd-closed set is invertible [Linear Algebra Appl., 174 (1992), pp. 65--74]. The aim of this note is to show that this conjecture holds if $n\le 7$, but it does not hold in general when $n\ge 8$.