A　complex　square　matrix　is　a　ray　pattern　matrix　if　each　of　its　nonzero　entries　has　modulus　1.　A　ray　pattern　matrix　naturally　corresponds　to　a　weighted-digraph.　A　ray　pattern　matrix　A　is　ray　nonsingular　if　for　each　entry-wise　positive　matrix　K,　A　omicron　K　is　nonsingular.　A　random　model　of　ray　pattern　matrices　with　order　n　is　introduced,　where　a　uniformly　random　ray　pattern　matrix　B　is　defined　to　be　the　adjacency　matrix　of　a　simple　random　digraph　D-n,D-　p　whose　arcs　are　weighted　with　i.i.d.　random　variables　uniformly　distributed　over　the　unit　circle　in　the　complex　plane.　It　is　shown　that　p*　(n)　=　1/n　is　a　threshold　function　for　the　random　matrix　M　=　I-B　to　be　ray　nonsingular,　which　is　the　same　as　the　threshold　function　of　the　appearance　of　giant　strong　components　in　D-n,D-p.