Bijective matrix algebra

If A and B are square matrices such that AB = I, then BA = I automatically follows. We prove a combinatorial version of this result in the case where the entries of A and B count collections of signed, weighted objects. Specifically, we give an algorithm that transforms any given bijective proof of the identity AB = I into an explicit bijective proof of the identity BA = I. Letting A and B be the Kostka matrix and its inverse, this settles an open problem posed by Eğecioğlu and Remmel in 1990.