is an odd number, because the circuits in such a matroid have size
Bipartite matroids were defined by Welsh (1969) as a generalization of the bipartite graphs, graphs in which every cycle has even size.
[1] An Eulerian graph is one in which all vertices have even degree; Eulerian graphs may be disconnected.
For planar graphs, the properties of being bipartite and Eulerian are dual: a planar graph is bipartite if and only if its dual graph is Eulerian.
For instance, the uniform matroid
It is possible to test in polynomial time whether a given binary matroid is bipartite.
[2] However, any algorithm that tests whether a given matroid is Eulerian, given access to the matroid via an independence oracle, must perform an exponential number of oracle queries, and therefore cannot take polynomial time.