A note on solutions of the matrix equation AXB=C

This paper deals with necessary and sufficient condition for consistency of the matrix equation $AXB=C$. We will be concerned with the minimal number of free parameters in Penrose’s formula $X=A^{(}1)C{B}^{(}1)+Y-A^{(}1)AYB{B}^{(}1)$ for obtaining the general solution of the matrix equation and we will establish the relation between the minimal number of free parameters and the ranks of the matrices A and B. The solution is described in the terms of Rohde’s general form of the {1}-inverse of the matrices A and B. We will also use Kronecker product to transform the matrix equation $AXB=C$ into the linear system $(B^T \otimes A)vecX = vec C$.