DIVISIBILITY AMONG POWER GCD MATRICES AND POWER LCM MATRICES

Abstract For any integers x and y , let $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of x and y , respectively. Let $a,b$ and n be positive integers, and let $S=\{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. We denote by $(S^a)$ and $[S^a]$ the $n\times n$ matrices having the a th power of $(x_i,x_j)$ and $[x_i,x_j]$ , respectively, as the $(i,j)$ -entry. Bourque and Ligh [‘On GCD and LCM matrices’, Linear Algebra Appl. 174 (1992), 65–74] showed that if S is factor closed (that is, S contains all positive divisors of any element of S ), then the GCD matrix $(S)$ divides the LCM matrix $[S]$ (written as $(S)\mid [S]$ ) in the ring $M_n({\mathbb Z})$ of $n\times n$ matrices over the integers. Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl. 428 (2008), 1001–1008] proved that $(S^a)\mid (S^b)$ , $(S^a)\mid [S^b]$ and $[S^a]\mid [S^b]$ in the ring $M_{n}({\mathbb Z})$ when $a\mid b$ and S is a divi