You can do pretty much anything you like. But I find it simplest to work from the top row to the bottom.
Since one thing you want to do to "row-reduce" a matrix is get "0" below the diagonal, each time you start on a row you will already have "0" to the left and row operations won't change that. If the "pivot" element- the nth column of the nth row so on the diagonal- is "0", you will want to swap it with a row below that has a non-zero number in that column. If you can't do that, if all numbers in that column below are also 0, then that matrix is NOT invertible, it has a 0 eigenvalue, and its row-reduced form has "0"s on the diagonal.
