The Smith normal form and left good matrix have been known in matrix theorem. Any matrix over the principal ideal ring has a Smith normal form. The Smith normal form of a matrix has many applications on various fields such as a solution of Diophantin linear equation and differential equation system. Furthermore, a matrix A with entries in a commutative ring R with unity is left good if for every vector x, the ideal Ã¡xAñ is the same as the ideal Ã¡Añ. This paper discusses the relation between the Smith normal form and left good matrix. The relation is as the following: matrix A with entries in principal ideal ring of size m by n, with

m < n, has Smith normal form [I_m, O] if only if A is a left good matrix.