Matrix representations of frame and lifted-graphic matroids correspond to gain functions

Let M be a 3-connected matroid and let F field. A matrix over representing (G,B) biased graph M. We characterize the relationship between (G,B), settling four conjectures of Zaslavsky. show that for each representation M, is projectively equivalent to canonical arising from G as gain F+ or F× realizing B. Further, we projective equivalence classes representations are in one-to-one correspondence with switching graphs except one degenerate case.