A matroid is defined to be a family of subsets of a finite set, satisfying certain axioms.
[6] The number of bases in a regular matroid may be computed as the determinant of an associated matrix, generalizing Kirchhoff's matrix-tree theorem for graphic matroids.
(the four-point line) is not regular: it cannot be realized over the two-element finite field GF(2), so it is not a binary matroid, although it can be realized over all other fields.
The matroid of the Fano plane (a rank-three matroid in which seven of the triples of points are dependent) and its dual are also not regular: they can be realized over GF(2), and over all fields of characteristic two, but not over any other fields than those.
The converse is true: every matroid that is realizable over both of these two fields is regular.
The result follows from a forbidden minor characterization of the matroids realizable over these fields, part of a family of results codified by Rota's conjecture.
[10] The equivalence of regular matroids and unimodular matrices, and their characterization by forbidden minors, are deep results of W. T. Tutte, originally proved by him using the Tutte homotopy theorem.
[8] Gerards (1989) later published an alternative and simpler proof of the characterization of unimodular matrices by forbidden minors.