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　S1　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.　©　2017　Elsevier　Inc.