Abstract :
Let S= x1,…,xn be a set of n distinct positive integers. Let 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 gcd-closed if (xi,xj) S for all 1 i,j n. For an integer x, let ν(x) denote the number of distinct prime factors of x. In this paper, by using the concept of greatest-type divisor introduced by S. Hong in [Adv. Math. (China) 25 (1996) 566–568; J. Algebra 218 (1999) 216–228], we obtain a new reduced formula for detf[(xi,xj)] if S is gcd-closed. Then we show that if S= x1,…,xn is a gcd-closed set satisfying maxx S ν(x) 2, and if f is a strictly increasing (respectively decreasing) completely multiplicative function, or if f is a strictly decreasing (respectively increasing) completely multiplicative function satisfying (respectively f(p) p) for any prime p, then the matrix [f(xi,xj)] (respectively (f[xi,xj])) defined on S is nonsingular. As a corollary, we show the following interesting result: The LCM matrix ([xi,xj]) defined on a gcd-closed set is nonsingular if maxx S ν(x) 2.
