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The generalized reflexive solutions of the matrix equation AX = B

In this paper, we consider the generalized reflexive solutions of the matrix equation AX = B. The necessary and sufficient conditions for the solvability of the problem are given, moreover, a simple and eigenvector-free formula of the general solution to the matrix equation is presented using Moore-Penrose generalized inverses of the coefficient matrix A. In addition, in corresponding solution set of the equation, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm has been provided.