The basic properties of the one-electron density matrix of a crystal are considered. It is shown that when the Brillouin zone special-point technique, developed earlier for calculating the electron density and local exchange potentials, is directly applied to the case of a nonlocal exchange potential, the calculated density matrix is not idempotent and physically meaningless divergences appear. To surmount these difficulties, a scheme is developed for interpolating the density matrix over the Brillouin zone in reciprocal space. A modification of the Hartree-Fock method for an infinite crystal is proposed in which the equations of the cyclic-cluster model are satisfied automatically. The electronic structures of perfect crystals of BN_hex, silicon, and futile are calculated using the Hartree-Fock method and the density-functional theory.