Approximating the inverse of a diagonally dominant matrix with positive elements

For an $n\times n$ diagonally dominant matrix $T=(t_{i,j})_{n\times n}$ with positive elements satisfying certain bounding conditions, we propose to use a diagonal matrix $S=(s_{i,j})_{n\times n}$ to approximate the inverse of $T$, where $s_{i,j}=\delta_{i,j}/t_{i,i}$ and $\delta_{i,j}$ is the Kronecker delta function. We derive an explicitly upper bound on the approximation error, which is in the magnitude of $O(n^{-2})$. It shows that $S$ is a very good approximation to $T^{-1}$.