  We present a new Fermi gas formalism for the ABJ matrix model. This
formulation identifies the effect of the fractional M2-brane in the ABJ matrix
model as that of a composite Wilson loop operator in the corresponding ABJM
matrix model. Using this formalism, we study the phase part of the ABJ
partition function numerically and find a simple expression for it. We further
compute a few exact values of the partition function at some coupling
constants. Fitting these exact values against the expected form of the grand
potential, we can determine the grand potential with exact coefficients. The
results at various coupling constants enable us to conjecture an explicit form
of the grand potential for general coupling constants. The part of the
conjectured grand potential from the perturbative sum, worldsheet instantons
and bound states is regarded as a natural generalization of that in the ABJM
matrix model, though the membrane instanton part contains a new contribution.
