  We consider the problem of determining rank loss conditions for a
concatenation of full-rank matrices, such that each row of the composing
matrices is scaled by a random coefficient. This problem has applications in
wireless interference management and recommendation systems. We determine
necessary and sufficient conditions for the design of each matrix, such that
the random ensemble will almost surely lose rank by a certain amount. The
result is proved by converting the problem to determining rank loss conditions
for the union of some specific matroids, and then using tools from matroid and
graph theories to derive the necessary and sufficient conditions. As an
application, we discuss how this result can be applied to the problem of
topological interference management, and characterize the linear symmetric
degrees of freedom for a class of network topologies.
