On dual of the generalized splitting matroids

Given a binary matroid M and a subset T⊆E(M), Luis A. Goddyn posed a problem that the dual of the splitting of M, i.e., ((MT)∗) is not always equal to the splitting of the dual of M, ((M∗)T). This persuade us to ask if we can characterize those binary matroids for which (MT)∗=(M∗)T. Santosh B. Dhotre answered this question for a two-element subset T. In this paper, we generalize his result for any subset T⊆E(M) and exhibit a criterion for a binary matroid M and subsets T for which (MT)∗ and (M∗)T are the equal. We also show that there is no subset T⊆E(M) for which, the dual of element splitting of M, i.e., ((M′T)∗) equals to the element splitting of the dual of M, ((M∗)′T).