$M$-matrix is a kind of special matrix which has wide applications. Many problems in biology, physics and social science and so on have close connection with $M$-matrice, hence researches on $M$-matrix is valuable. In this paper, firstly, some new notations are introduced, and two inequalities of element relation on strictly diagonally dominant $M$-matrix and the inverse matrix are given. Secondly, some new upper bounds for the infinity norm of inverse matrix are obtained. Finally, the lower bound of the smallest eigenvalue of matrix $A$ is presented, which only depends on the elements of matrix $A$. The theoreical analysis and numerical examples show that the new upper bounds improve the related results.