Characterization for the general solution to a system of matrix equations with quadruple variables

In this paper, we give some necessary and sufficient conditions for the solvability to the system of matrix equationsequation(0.1)A1X1=C1,X2B1=D1A2X3=C2,X3B2=D2,A3X4=C3,X4B3=D3,A4X1+X2B4+C4X3D4+C5X4D5=E1and provide an expression of the general solution to (0.1). Furthermore, we obtain the maximal and minimal ranks of X3X3 and X4X4 in (0.1). The findings of this paper extend the known results in the literatures.