The author shows that if \(S=\{x_1, x_2,\dots, x_n\}\) is a multiple closed set or a divisor chain of positive integers and if \(f\) is a completely multiplicative function satisfying certain conditions, then the GCD matrix \((f((x_i, x_j)))\) divides the LCM matrix \((f([x_i, x_j]))\) in the ring \(M_n(\mathbb{Z})\) of \(n\times n\) matrices over the integers. The author also notes that this result does not hold for multiplicative functions. For related results, see K. Bourque and S. Ligh [Linear Algebra Appl. 174, 65–74 (1992; Zbl 0761.15013), ibid. 216, 267–275 (1995; Zbl 0826.15013)] and S. Hong [ibid. 345, 225–233 (2002; Zbl 0995.15006), Southeast Asian Bull. Math. 27, No. 4, 615–621 (2003; Zbl 1160.11316)]. The inverse formula for \((f((x_i, x_j)))\) on multiple closed sets presented in this paper is given in a more general setting in I. Korkee and P. Haukkanen [Linear Algebra Appl. 372, 127–153 (2003; Zbl 1036.06005)].