Let S = { x 1 , x 2 , … , x n } be a set of positive integers, and let f be an arithmetical function. The matrices ( S ) f = [ f ( gcd ( x i , x j ) ) ] and [ S ] f = [ f ( lcm   [ x i , x j ] ) ] 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 M n ( ℤ ) of the n × n matrices over the integers.