In　this　paper,　the　concept　of　determinants　for　the　matrices　over　a　commutative　semiring　is　introduced,　and　a　development　of　determinantal　identities　is　presented.　This　includes　a　generalization　of　the　Laplace　and　Binet-Cauchy　Theorems,　as　well　as　on　adjoint　matrices.　Also,　the　determinants　and　the　adjoint　matrices　over　a　commutative　difference-ordered　semiring　are　discussed　and　some　inequalities　for　the　determinants　and　for　the　adjoint　matrices　are　obtained.　The　main　results　in　this　paper　generalize　the　corresponding　results　for　matrices　over　commutative　rings,　for　fuzzy　matrices,　for　lattice　matrices　and　for　incline　matrices.　©　2013　Taylor　&　Francis.