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In 1876 H. J. S. Smith defined an LCM matrix as follows: let S = {x_1, x_2, ..., x_n} be a set of positive integers. The LCM matrix [S] is the n $\times$ n matrix with lcm(x_i , x_j) as its ij entry. During the last 30 years singularity of LCM matrices has interested many authors. In 1992 Bourque and Ligh ended up conjecturing that if the GCD close...
Let S={x1,x2,…,xn} be a finite set of distinct positive integers. Throughout this article we assume that the set S is GCD closed. The LCM matrix [S] of the set S is defined to be the n×n matrix with lcm(xi,xj) as its ij element. The famous Bourque-Ligh conjecture used to state that the LCM matrix of a GCD closed set S is always invertible, but curr...
Let $S=\{x_1,x_2,\ldots,x_n\}$ be a finite subset of distinct positive integers. Throughout this article we also assume that our set $S$ is GCD closed. The LCM matrix $[S]$ of the set $S$ is defined to be the $n\times n$ matrix with $\mathrm{lcm}(x_i,x_j)$ as its $ij$ element. The famous Bourque-Ligh conjecture used to state that the LCM matrix of...
The invertibility of LCM matrices and their Hadamard powers have been studied
a lot over the years by many authors. Bourque and Ligh conjectured in 1992 that
the LCM matrix $[S]=[[x_i, x_j]]$ on any GCD closed set $S=\{x_1, x_2, \ldots,
x_n\}$ is invertible, but in 1997 this was proven false. However, currently
there are many open conjectures conce...