Let a , b and n be positive integers with n ≥ 2 , f be an integer-valued arithmetic function, and the set S = { x 1 , ... , x n } of n distinct positive integers be a divisor chain such that x 1 | x 2 | ⋯ | x n. We first show that the matrix (f a (S)) having f evaluated at the a th power (x i , x j) a of the greatest common divisor of x i and x j as its i , j -entry divides the GCD matrix (f b (S)) in the ring M n (Z) of n × n matrices over integers if and only if f b − a (x 1) ∈ Z and (f a (x i) − f a (x i − 1)) divides (f b (x i) − f b (x i − 1)) for any integer i with 2 ≤ i ≤ n. Consequently, we show that the matrix (f a [ S ]) having f evaluated at the a th power [ x i , x j ] a of the least common multiple of x i and x j as its i , j -entry divides the matrix (f b [ S ]) in the ring M n (Z) if and only if f b − a (x n) ∈ Z and (f a (x i) − f a (x i − 1)) divides (f b (x i) − f b (x i − 1)) for any integer i with 2 ≤ i ≤ n. Finally, we prove that the matrix (f a (S)) divides the matrix (f b [ S ]) in the ring M n (Z) if and only if f a (x 1) | f b (x i) and (f a (x i) − f a (x i − 1)) | (f b (x i) − f b (x i − 1)) for any integer i with 2 ≤ i ≤ n. Our results extend and strengthen the theorems of Hong obtained in 2008. [ABSTRACT FROM AUTHOR]