Syzygy and Adjoint Matroids

We give a matroid-theoretic analogue of Gaussian elimination and use it to construct, for every matroid M, the syzygy matroid Syz(M) on the collection of circuits of M. Dualizing the construction yields the combinatorial adjoint Adj(M) on the collection of maximal proper flats of M, and the two are related by Adj(M) \simeq Syz(M*). Neither construction depends on a realization. We determine Syz(M) when M is U_{k,k+3} or the Vamos matroid V_{8}. We also show that Adj(M) is an adjoint of M when M has rank 3. When M has rank 4 and V_{8} is a restriction of M, Adj(M) has rank 3 and is not an adjoint of M.