The general solutions to some systems of matrix equations

In this paper, we consider an expression of the general solution to the classical system of matrix equations A(11)X B-11 = C-11, A(22)X B-22 = C-22, A(33)X B-33 = C-33. We present a necessary and sufficient condition for the existence of a solution to the system by using generalized inverses. We give an expression of the general solution to the system when it is solvable. As applications, we derive some necessary and sufficient conditions for the consistence to the system A(11) X A(11)* = C-11, A(22)X A(22)* = C-22, A(33)X A(33)* = C-33, X = X*, and the system A(11)X A(11)* = C-11, A(22)X B-22 = C-22, X = X*, where * means conjugate transpose. We also give the expressions of the general solutions to the systems.