  We construct an intersection product on tropical cycles contained in the
Bergman fan of a matroid. To do this we first establish a connection between
the operations of deletion and restriction in matroid theory and tropical
modifications as defined by Mikhalkin. This product generalises the product of
Allermann and Rau, and Allermann and also provides an alternative procedure for
intersecting cycles which is not based on intersecting with Cartier divisors.
Also, we simplify the definition in the case of one dimensional fan cycles in
two dimensional matroidal fans and given an application of the intersection
product to realisability questions in tropical geometry.
