The situation and other variations of the algorithm

If the equations are and the matrix needs a permutation of rows, then we can first apply it to get new equations and the use the same procedure as above with the new right hand side .

The matrix is not necessarily unique and indeed sometimes, we make row permutations for reasons other than zero pivots. For example, it is a principle of numerical analysis that better numerical stability/accuracy results if we choose a larger pivot than a smaller one. So, sometimes a row can be swapped just to get a bigger pivot.
Sometimes, especially in hand calculations, we may prefer a pivot rather than another larger number, so we might swap rows.

Sometimes, we may wish to avoid all fractions and the elimination process is replaced by a much slower specialized process where we don't try to make entries zero immediately in one step, but use steps to reduce the entries until they become zero. The process resembles the calculation of GCD in number theory or algebra. Then the process has a different use besides just getting a solution. We show such a sample calculation as an illustration. As before, we augment the identity matrix to keep track of the total transformation.

> F:=matrix(2,2,[3,4,2,3]);FF:=augment(F,eye(2));

> FF1:=addrow(FF,1,2,-1);

> FF2:=addrow(FF1,2,1,3);

> FF3:=swaprow(FF2,1,2);

> FF4:=multrow(FF3,1,-1);

> FF5:=addrow(FF4,2,1,-1);

>

The main reasons to mention the above is that you should not get the impression that there is only one way to solve the equations . You can and should experiment with alternate methods suited to the requirements.