  In ArXiv:1604.00338[math.QA] we gave a complete combinatorial
characterization of homogeneous quadratic identities for minors of quantum
matrices. It was obtained as a consequence of results on minors of matrices of
a special sort, the so-called path matrices $Path_G$ generated by paths in
special planar directed graphs $G$.
  In this paper we prove two assertions that were stated but left unproved in
ArXiv:1604.00338[math.QA]. The first one says that any minor of $Path_G$ is
determined by a system of disjoint paths, called a flow, in $G$ (generalizing a
similar result of Lindstr\"om's type for the path matrices of Cauchon graphs by
Casteels). The second, more sophisticated, assertion concerns certain
transformations of pairs of flows in $G$.
