A　uniformly　random　ray　pattern　matrix　A　with　a　given　zero　nonzero　pattern　(described　by　a　digraph　D　with　no　multi-arcs　or　loops)　is　the　matrix　whose　nonzero　entries　are　mutually　independent　random　variables　uniformly　distributed　over　the　unit　circle　S-1　in　the　complex　plane.　It　is　shown　in　this　paper　that　the　probability　of　I　-　A　to　be　ray　nonsingular　is　completely　determined　by　the　cycle　graph　cg(D)　of　D　(i.e.　the　adjacency　structure　of　the　directed　cycles　in　D)　if　cg(D)　is　a　tree.　A　formula　is　given　to　compute　the　probability　when　cg(D)　is　a　tree,　and　it　is　also　shown　that　as　the　order　of　cg(D)　tends　to　infinity,　the　limit　of　the　probability　is　0.　(C)　2017　Elsevier　Inc.　All　rights　reserved.