Making a diagonally dominant matrix sparse

I need to solve a system of the sort: Ax=b where the unknown is the matrix A. This matrix is symmetric and many times for smaller inputs happened to be diagonally dominant https://en.wikipedia.org/wiki/Diagonally_dominant_matrix. For the moment I regard it a dense matrix and thus get out of memory as n^2 gets out of hand fast. Any idea on how to tackle the problem?