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Presented in this paper is a new matrix representation of classical electromagnetic theory. The basis of this representation is a space-time, eight- by-eight differential matrix operator. This matrix operator is initially formulated from the differential form of the Maxwell field equations in vacuum. The resulting matrix formulation of Maxwell's equations allows simple and direct derivation of the electromagnetic wave and charge continuity equation s, the Lorentz conditions and definition of the electromagnetic potentials, the Lorentz and Coulomb gauges, the electromagnetic potential wave equations, and Poynting's conservation of energy theorem. A four-dimensional Fourier transform of the matrix equations casts them into an eight-dimensional transfer theorem. The transfer function has an inverse, and this allows the equations to be inverted. This inversion expresses the fields directly in terms of the charge and current source distributions, i. e., without the need for calculating intermediary potentials. This inversion formula is new, for the general scenario of space- and time-dependent sources. A simple pedagogical example is included illustrating use of the formulation.