Notes on Hong's conjecture on nonsingularity of power LCM matrices

Let $a,n$ be positive integers and $S=\{x_1,...,x_n\}$ be a set of $n$ distinct positive integers. The set $S$ is said to be gcd (resp. lcm) closed if $gcd(x_i,x_j)\in S$ (resp. $[x_i,x_j]\in S$) for all integers $i,j$ with $1\le i,j\le n$. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of the greatest common divisor (resp. the least common multiple) of $x_i$ and $x_j$ as its $(i,j)$-entry. In this paper, we mainly show that for any positive integer $a$ with $a\ge 2$, the power LCM matrix $[S^a]$ defined on a certain class of gcd-closed (resp. lcm-closed) sets $S$ is nonsingular. This provides evidences to a conjecture raised by Shaofang Hong in 2002.