An　antiring　is　a　semiring　which　is　zerosumfree　(i.e.,　a　+　b　=　0　implies　a　=　b　=　0　for　any　a,　b　in　this　semiring).　In　this　paper,　the　complete　description　of　the　invertible　matrices　over　a　commutative　antiring　is　given　and　some　necessary　and　sufficient　conditions　for　a　matrix　over　a　commutative　antiring　to　be　invertible　are　obtained.　Also,　Cramer＇s　rule　over　commutative　antirings　is　presented.　The　main　results　in　this　paper　generalize　and　develop　the　corresponding　results　for　the　Boolean　matrices,　the　fuzzy　matrices,　the　lattice　matrices　and　the　incline　matrices.　©　2007　Elsevier　Inc.　All　rights　reserved.