Divisibility properties of power GCD matrices and power LCM matrices Original Research Article

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 ab, 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 adoes not divideb and ngreater-or-equal, slanted2, 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.