The GCD-reciprocal LCM matrices on GCD-closed sets (Q880464)

Let \(S=\{x_1,x_2,\dots,x_n\}\) be a set of positive integers, and let \(A\) denote the \(n\times n\) matrix whose \((i, j)\)-entry is \({\text{ gcd}(x_i, x_j)\over \text{ lcm}[x_i, x_j]}\). The authors evaluate the determinant and the inverse of the matrix \(A\) on gcd-closed sets \(S\). \textit{I. Korkee} [JP J. Algebra Number Theory Appl. 5, No. 1, 75--88 (2005; Zbl 1083.15013)] has studied this problem in a much more general level. These papers belong to the field of Smith's determinant, see, e.g., \textit{P. Haukkanen, J. Wang} and \textit{J. Sillanpää} [Linear Algebra Appl. 258, 251--269 (1997; Zbl 0883.15002)].