International Journal of Mathematics and Mathematical Sciences 
Volume 2005 (2005), Issue 6, Pages 925-935
doi:10.1155/IJMMS.2005.925

Notes on the divisibility of GCD and LCM Matrices

Pentti Haukkanen and Ismo Korkee

 Department of Mathematics, Statistics and Philosophy, University of Tampere, Tampere 33014, Finland
Received 10 November 2004

Abstract

Let S={x1,x2,…,xn} be a set of positive integers, and let f be an arithmetical function. The matrices (S)f=[f(gcd(xi,xj))] and [S]f=[f(lcm [xi,xj])] are referred to as the greatest common divisor (GCD) and the least common multiple (LCM) matrices on S with respect to f, respectively. In this paper, we assume that the elements of the matrices (S)f and [S]f are integers and study the divisibility of GCD and LCM matrices and their unitary analogues in the ring Mn(ℤ) of the n×n matrices over the integers.