If \(f\) is an arithmetic function, and \(S=\{x_1<x_2<\dots<x_n\}\) is a set of integers, then one can consider the matrices \((S)_f=[f(\text{GCD}(x_i,x_j))]\) and \([S]_f=[f(\text{LCM}(x_i,x_j))]\). Divisibility properties of these matrices have been considered by several authors [see e.g. K. Bourque and S. Ligh, Lin. Algebra Appl. 174, 65–74 (1992; Zbl 0761.15013), Lin. Multilin. Algebra 34, 261–267 (1993; Zbl 0815.15022); S. Hong, Lin. Algebra Appl. 345, 225–233 (2002; Zbl 0995.15006), Colloq. Math. 98, 113–123 (2003; Zbl 1047.11023)].
The authors present several generalization of these results, considering in particular the unitary analogues of \((S)_f\) and \([S]_f\). They also give counterexamples to two conjectures stated recently by S. Hong [Acta Arith. 111, 165–177 (2004; Zbl 1047.11022)].