Let me take this opportunity to make a comment that talks about the upper bound, not the lower.

Actually, let me remind you that it is a common false belief that Gaussian Elimination has $O(n^3)$ complexity. See the nice question with its answers [here at cstheory][1]

This misbelief happens because even though GE requires $O(n^3)$ arithmetic operations, if not done properly, there can be massive intermediate coefficient growth, which renders judging the true complexity of GE a difficult task.

You might also enjoy [looking here][2] at the near linear time algorithms for solving special linear systems (example for diagonally dominant matrices). For general, non-structured matrices, the situation is less clear.


  [1]: https://cstheory.stackexchange.com/questions/3921/what-is-the-actual-time-complexity-of-gaussian-elimination
  [2]: http://cs-www.cs.yale.edu/homes/spielman/precon/precon.html