  A chiral matrix model applicable to QCD with 2+1 flavors is developed. This
requires adding a SU(3)_L x SU(3)_R x Z(3)_A nonet of scalar fields, with both
parities, and coupling these to quarks through a Yukawa coupling, y. Treating
the scalar fields in mean field approximation, the effective Lagrangian is
computed by integrating out quarks to one loop order. In addition to the usual
symmetry breaking term, linear in the current quark mass m_qk, at a nonzero
temperature T it is necessary to add a new term, ~ m_qk T^2. The parameters of
the gluon part of the matrix model are identical to that for the pure glue
theory without quarks. The parameters in the chiral matrix model are fixed by
the values, at zero temperature, of the pion decay constant and the masses of
the pions, kaons, eta, and eta'. The temperature for the chiral crossover at
T_chi = 155 MeV is determined by adjusting the Yukawa coupling y. We find
reasonable agreement with the results of numerical simulations on the lattice
for the pressure and related quantities. In the chiral limit, besides the
divergence in the chiral susceptibility there is also a milder divergence in
the susceptibility between the Polyakov loop and the chiral order parameter,
with critical exponent beta - 1. We compute derivatives with respect to a quark
chemical potential to determine the susceptibilities for baryon number, the
chi_2n. Especially sensitive tests are provided by chi_4 - chi_2 and by chi_6,
which changes in sign about T_chi. The behavior of the susceptibilities in the
chiral matrix model strongly suggests that as the temperature increases from
T_chi, that the transition to deconfinement is significantly quicker than
indicated by the measurements of the (renormalized) Polyakov loop on the
lattice.
