  We study matroidal networks introduced by Dougherty et al. We prove the
converse of the following theorem: If a network is scalar-linearly solvable
over some finite field, then the network is a matroidal network associated with
a representable matroid over a finite field. It follows that a network is
scalar-linearly solvable if and only if the network is a matroidal network
associated with a representable matroid over a finite field. We note that this
result combined with the construction method due to Dougherty et al. gives a
method for generating scalar-linearly solvable networks. Using the converse
implicitly, we demonstrate scalar-linear solvability of two classes of
matroidal networks: networks constructed from uniform matroids and those
constructed from graphic matroids.
