  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 concerning LCM matrices and their Hadamard
powers presented by Hong. In this paper we utilize lattice-theoretic structures
and the M\"obius function to explain the singularity of classical LCM matrices
and their Hadamard powers. At the same time we end up disproving some of Hong's
conjectures. Elementary mathematical analysis is applied to prove that for most
semilattice structures there exist a set $S=\{x_1, x_2, \ldots, x_n\}$ of
positive integers and a real number $\alpha>0$ such that $S$ possesses this
structure and the power LCM matrix $[[x_i, x_j]^\alpha]$ is singular.
