Classification and compatibility of energy-momentum tensors

We present, in detail, the Segre classification of the energy-momentum tensors associated with: (a) a fluid with anisotropic pressure and heat flux, and (b) a perfect fluid with an electromagnetic field. These are the most widely used energy-momentum tensors in general relativity. We discuss algebraic consistency and how it may be used to simplify the process of obtaining exact solutions to Einstein's field equations. Further, we shed considerable light on the meaning of syzygies amongst the Riemann invariants.