@article{Zhu2022, 
author = {Guangyan Zhu and Kaimin Cheng and Wei Zhao},
title = {Notes on Hong's conjecture on nonsingularity of power LCM matrices},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {6},
pages = {10276-10285},
keywords = {power LCM matrix, power GCD matrix, nonsingularity, gcd-closed set},
url = {https://www.sciopen.com/article/10.3934/math.2022572},
doi = {10.3934/math.2022572},
abstract = {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    )  ∈  S (resp.    [      x    i    ,      x    j    ]  ∈  S) for all integers    i  ,  j with    1  ≤  i  ,  j  ≤  n. We denote by    (      S    a    ) (resp.    [      S    a    ]) the    n  ×  n matrix having the    ath 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  ≥  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.}
}