  We investigate a gauged matrix model in the large $N$ limit which is closely
related to the superconductor fluctuation and the flux lattice melting in two
dimensions. With the use of saddle point method the free energy is expanded up
to eighth order for the coupling constant $g$. In the case that the coefficient
of quadratic term of the Ginzburg-Landau matrix model is negative, a critical
point $g=g_c$ is obtained in the large $N$ limit and the relation between this
phase transition and the 2D flux lattice melting transition is discussed.
