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.