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Let n n complex matrices P andQ be nontrivial generalized reflection matrices, i.e., P D P D P 1 ¤ In, Q DQ DQ 1 ¤ In. A complex matrix A with order n is said to be a .P;Q/ generalized anti-reflexive matrix, if PAQ D A. We in this paper mainly investigate the .P;Q/ generalized anti-reflexive maximal and minimal rank solutions to the system of matrix equation AX D B . We present necessary and sufficient conditions for the existence of the maximal and minimal rank solutions, with .P;Q/ generalized anti-reflexive, of the system. Expressions of such solutions to this system are also given when the solvability conditions are satisfied. In addition, in correspondence with the minimal rank solution set to the system, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm has been provided. 2000 Mathematics Subject Classification: 15A29

The $(P,Q)$ generalized anti-reflexive extremal rank solutions to a system of matrix equations