A　complex　square　matrix　is　called　a　ray　nonsingular　matrix　(RNS　matrix)　if　its　ray　pattern　implies　that　it　is　nonsingular.　A　matrix　M　=　I　-A(W)　is　called　a　cycle　tree　matrix　if　the　adjacency　structure　of　the　cycles　in　the　arc-weighted　digraph　W　(with　no　multi-arcs　or　loops),　which　is　described　by　the　cycle　graph　of　W,　is　a　tree.　In　this　paper,　it　is　shown　that　if　there　is　no　positive　cycle　in　W,　then　the　cycle　tree　matrix　M　=　I　-　A(W)　is　a　forbidden　structure　for　RNS　if　and　only　if　M　is　not　RNS.　(C)　2020　Elsevier　B.V.　All　rights　reserved.