On the Hadamard products of GCD and LCM matrices

Abstract

Let S = {xı, X2,..., Xn} be a set of distinct positive integers. The matrix (S) having the greatest common divisor (xi , Xj) of Xi and Xj as its i, j-entry is called the greatest common divisor (GCD) matrix on S. The matrix [S] having the least common multiple [xi, Xj] of Xi and Xi as its i, j- entry is called the least common multiple (LCM) matrix on S. In this paper we obtain some results related with Hadamard products of GCD and LCM matrices. The set S is factor-closed if it contains every divisor of each of its elements. It is well-known,that if S is factor-closed,then there exit the inverses of the GCD and LCM matrices on S. So we conjecture that if the set S is factor-closed, then (S)o(S)’‘ and [S]o[S] ' n matrices are doubly stochastic matrices and tr((S)o(S)-')=lı((S))= £ Xi-