  Consider an ample and globally generated line bundle $L$ on a smooth
projective variety $X$ of dimension $N\geq 2$ over $\mathbb{C}$. Let $D$ be a
smooth divisor in the complete linear system of $L$. We construct reflexive
sheaves on $X$ by an elementary transformation of a trivial bundle on $X$ along
certain globally generated torsion-free sheaves on $D$. The dual reflexive
sheaves are called the Lazarsfeld-Mukai reflexive sheaves. We prove the
$\mu_L$-(semi)stability of such reflexive sheaves under certain conditions.
