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Some necessary and sufficient conditions for the existence of the   ?-skew-Hermitian solution quaternion matrix equations the system of matrix   equations with ?-skew-Hermicity, A1X = C1, XB1 = C2, A2Y = C3, YB2 = C4, X =   -X?*; Y=-Y?*, A3XA?*3 + B3YB?*3=C5, are established in this paper by using rank equalities of the coefficient matrices. The general solutions to   the system and its special cases are provided when they are consistent. Within the framework of the theory of noncommutative row-column   determinants, we also give determinantal representation formulas of finding   their exact solutions that are analogs of Cramer?s rule. A numerical example   is also given to demonstrate the main results.

Explicit formulas and determinantal representation for η-skew-Hermitian solution to a system of quaternion matrix equations