Notes on Hong's conjectures of real number power LCM matrices

Let e be a real number and S={x1,…,xn} be a set of n distinct positive integers. The set S is said to be gcd-closed (respectively lcm-closed) if (xi,xj)∈S (respectively [xi,xj]∈S) for all 1⩽i,j⩽n. The matrix having eth power e[xi,xj] of the least common multiple of xi and xj as its i,j-entry is called the eth power least common multiple (LCM) matrix, denoted by (e[xi,xj]) (or abbreviated by (e[S])). In this paper, we show that for any real number e⩾1 and n⩽7, the power LCM matrix (e[xi,xj]) defined on any gcd-closed (respectively lcm-closed) set S={x1,…,xn} is nonsingular. This confirms partially two conjectures raised by Hong in [S. Hong, Nonsingularity of matrices associated with classes of arithmetical functions, J. Algebra 281 (2004) 1–14]. Similar results are established for reciprocal real number power GCD matrices.