Classification and compatibility of energy-momentum tensors

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