On a generalization of the reciprocal LCM matrix

Abstract

Let S = {x,,x2,...,x,} be a set of distinct positive integers. The matrix 1 /[S] = (s .), where sa = l/[x„X j], the reciprocal of the least common multiple of x, and x ,, is called the reciprocal least common multiple (reciprocal LCM) matrix on S . In this paper, we present a generalization of the reciprocal LCM m atrixon S , that is the matrix 1 /[£ '], the ij- entry ofw hichis l/[ x ,,x jr , where r is a real number. We obtain a structure theorem for l/[5 '] and the value of the determinant of 1 /[S' ]. We also prove that 1/[S'] is positive definite if r > 0 . Then we calculate the inverse of l/[5 r ] on a factor closed set. Finally, we show that the matrix [•$"] = ([ x ,,x j') defined on S is the product of an integral matrix and the generalized reciprocal LCM matrix l/[5,r ] = (l/[x ,,x j') if S is factor closed and r is a positive integer.