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A necessary and sufficient condition is given for the quaternion matrix equations
AiX + Y Bi = Ci (i = 1, 2) to have a pair of common solutions X and Y . As a consequence, the
results partially answer a question posed by Y.H. Liu (Y.H. Liu, Ranks of solutions of the linear
matrix equation AX + Y B = C, Comput. Math. Appl., 52 (2006), pp. 861-872).