Nonsingularity of matrices associated with classes of arithmetical functions on lcm-closed sets

Let S = {x1, … , xn} be a set of n distinct positive integers and f be an arithmetical function. Let [f(xi, xj)] denote the n × n matrix having f evaluated at the greatest common divisor (xi, xj) of xi and xj as its i, j-entry and (f[xi, xj]) denote the n × n matrix having f evaluated at the least common multiple [xi, xj] of xi and xj as its i, j-entry. The set S is said to be lcm-closed if [xi, xj] ∈ S for all 1 ⩽ i, j ⩽ n. For an integer x > 1, let ω(x) denote the number of distinct prime factors of x. Define ω(1) = 0. In this paper, we show that if S = {x1, … , xn} is an lcm-closed set satisfying , and if f is a strictly increasing (resp. decreasing) completely multiplicative function, or if f is a strictly decreasing (resp. increasing) completely multiplicative function satisfying (resp. f(p) ⩾ p) for any prime p, then the matrix [f(xi, xj)] (resp. (f[xi, xj])) defined on S is nonsingular. By using the concept of least-type multiple introduced in [S. Hong, J. Algebra 281 (2004) 1–14], we also obtain reduced formulas for det(f(xi, xj)) and det(f[xi, xj]) when f is completely multiplicative and S is lcm-closed. We also establish several results about the nonsingularity of LCM matrices and reciprocal GCD matrices.