  In this article, we construct a non-commutative crepant resolution (=NCCR) of
a minimal nilpotent orbit closure $\overline{B(1)}$ of type A, and study
relations between an NCCR and crepant resolutions $Y$ and $Y^+$ of
$\overline{B(1)}$. More precisely, we show that the NCCR is isomorphic to the
path algebra of the double Beilinson quiver with certain relations and we
reconstruct the crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$ as
moduli spaces of representations of the quiver. We also study the
Kawamata-Namikawa's derived equivalence between crepant resolutions $Y$ and
$Y^+$ of $\overline{B(1)}$ in terms of an NCCR. We also show that the P-twist
on the derived category of $Y$ corresponds to a certain operation of the NCCR,
which we call multi-mutation, and that a multi-mutation is a composition of
Iyama-Wemyss's mutations.
