A note on solutions of the matrix equation AXB=C

Abstract: 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 $X = A^1CB^1 + Y - A^1AYBB^1$ 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 $(B^T \otimes A)vecX = vec C$.