  An analytical expression for the first-order density matrix of a charged,
two-dimensional, harmonically confined quantum gas, in the presence of a
constant magnetic field is derived. In contrast to previous results available
in the literature, our expressions are exact for any temperature and magnetic
field strength. We also present a novel factorization of the Bloch density
matrix in the form of a simple product with a clean separation of the
zero-field and field-dependent parts. This factorization provides an
alternative way of analytically investigating the effects of the magnetic field
on the system, and also permits the extension of our analysis to other
dimensions, and/or anisotropic confinement.
