Divisibility properties of power GCD matrices and power LCM matrices

Let a,b and n be positive integers and the set S={x1,…,xn} of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on {1,…,n} such that xσ(1)|…|xσ(n)). In this paper, we show that if a|b, then the ath power GCD matrix (Sa) having the ath power (xi,xj)a of the greatest common divisor of xi and xj as its i,j-entry divides the bth power GCD matrix (Sb) in the ring Mn(Z) of n×n matrices over integers. We show also that if a∤b and n⩾2, then the ath power GCD matrix (Sa) does not divide the bth power GCD matrix (Sb) in the ring Mn(Z). Similar results are also established for the power LCM matrices.