Ray　nonsingular　(RNS)　matrices　are　a　generalization　of　sign　nonsingular　(SNS)　matrices　from　the　real　field　to　complex　field.　The　problem　per,　we　give　an　algorithm　to　recognize　the　ray　nonsingularity　for　the　so　called　＂cycle　chain　matrices＂,　whose　associated　digraphs　are　of　certain　simple　structure.　Furthermore,　the　cycle　chain　matrices　that　are　not　ray　nonsingular　are　classified　into　two　classes　according　to　whether　or　not　they　can　be　majorized　by　some　RNS　cycle　chain　matrices,　and　the　case　when　they　can　is　characterized.　(c)　2012　Elsevier　Inc.　All　rights　reserved.