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

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