We establish structural controllability results for matrix pairs [A, B] whereA=A0+ Σ μiAi,B=B0+ σ μi, Bi, with theAi,Bi, fixed, and the μi, free scalar parameters. The results characterize structural controllability in several ways, via tests involving the checking of the rank or the evaluation of the determinant of various constant matrices formed from theAi,Bi. A number of the results used as intermediate results tests for a full rank property of matrix nets, i.e. tests that check ifM=M0+ ΣμiMiprescribed, μivariable, has full rank for almost all μi.