Abstract: Based on the other earlier works as shown in reference and the characteristics of the elements of an irreducibly and weakly diagonally dominant matrix, the row elements of the complex matrix A are divided into three parts, then the module of the elements of each and every part are summed up to obtain the three values αi, βi, and γi. Comparing the magnitudes of the values of hik and Hjk, which are both constructed by αi, βi, and γi, the sufficient conditions are given to judge whether an irreducibly and weakly diagonally dominant matrix is a generalized strictly diagonally dominant matrix or not. Then, the results can be applied to the judgment of a nonsingular M-matrix so as to extend the main conclusions as indicated by other earlier works.