Solving a generalized characteristic matrix equation (GCME) of a linear retarded system is the crux for stabilizing the system using reduction technique. This note gives the solution of GCME under the general condition that the eigenvalues occur in the spectrum of the system with algebraic and geometric multiplicities being greater than or equal to one. The main idea is to transform the GCME into a group of linear algebraic equations. A sufficient condition for the existence of the solution to the linear algebraic equations is established. A key lemma for stabilizing the systems under the above general condition is also proved.