In this paper, Let the matrix $A\in C^{n\times n}$ with $Ind(A)=k$, we first construct two bordered matrices based on [32], which gave a method for computing the null space of $A^k$ by applying elementary row operations on the pair $\begin{pmatrix}A&I\end{pmatrix}$. Then two new Algorithms to compute the Drazin inverse $A^d$ are presented based on elementary row operations on two partitioned matrices. The computational complexities of the two Algorithms are detailed analyzed. When the index $k=Ind(A)\geq5$, the two Algorithms are all faster than the Algorithm by Anstreicher and Rothblum [32]. In the end, an example is presented to demonstrate the two new algorithms.