Row Echelon Form: Does Swapping Rows Change Matrix?

[jinksys]

Say I am given a matrix and am supposed to put the matrix in row echelon form, does swapping two rows change the final matrix? Say I have:

3 1 4
1 1 1
0 1 3

It would save time to swap the first two rows, however when I do that on my problems the first row is wrong.

 

[jinksys]

I've found my answer.

Seems that REFs aren't unique and depend on the order and number of EROs.

 

[Hurkyl]

Reduced row echelon forms should be unique, though.