  Let $G$ be a finite undirected multigraph with no self-loops. The Jacobian
$\operatorname{Jac}(G)$ is a finite abelian group associated with $G$ whose
cardinality is equal to the number of spanning trees of $G$. There are only a
finite number of biconnected graphs $G$ such that the exponent of
$\operatorname{Jac}(G)$ equals $2$ or $3$. The definition of a Jacobian can
also be extended to regular matroids as a generalization of graphs. We prove
that there are finitely many connected regular matroids $M$ such that
$\operatorname{Jac}(M)$ has exponent $2$ and characterize all such matroids.
