Proof of a conjecture of Hong on nonsingularity of power LCM matrices on gcd-closed sets

For any integers x x and y y , let ( x , y ) (x, y) and [ x , y ] [x, y] stand for the greatest common divisor and the least common multiple of x x and y y , respectively. Let e e and n n be positive integers. Let S = { x 1 , ⋯ , x n } S=\{x_1, ⋯ , x_n\} be a set of n n distinct positive integers. We denote by ( S e ) (S^e) and [ S e ] [S^e] the n × n n× n matrix having the e e th power of ( x