Explicit solutions to the matrix inverse problem AX = B

Recently, Wu and Cain gave necessary and sufficient conditions for the matrix equation AX = B to have a solution with respect to A subject to the condition that the Hermitian part of A is non-negative definite. In this note we give an easier approach, which also provides possible explicit expressions for A. Further, necessary and sufficient conditions are derived when the Hermitian part of A is prescribed to be positive definite, thus correcting a result of Wu and Cain.