Let \(M = (E, \mathcal {F})\) be a matroid on a set \(E\), \(B\) be one of its bases, and \(M_B\) the base matroid associated to \(B\). In this paper, the authors extend the notion of cycles intersecting in a path to matroids by defining the notion of when two circuits are \(p\)-intersecting. The authors prove that a simple binary matroid \(M\) is not isomorphic to \(M_B\) for any base \(B\) if and only if \(M\) contains a pair of \(p\)-intersecting circuits or a covered circuit.