Row Echelon Form: Does Swapping Rows Change Matrix?

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Swapping rows in a matrix does not change the final row echelon form (REF), as REF is not unique and can vary based on the order and number of elementary row operations (EROs) performed. In the provided example, swapping the first two rows could save time, but it may lead to confusion if not properly managed. However, reduced row echelon forms (RREFs) are unique regardless of the row operations used. Understanding the distinction between REF and RREF is crucial for accurate matrix manipulation. Ultimately, careful application of row operations is essential for achieving the desired matrix form.
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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.
 
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I've found my answer.

Seems that REFs aren't unique and depend on the order and number of EROs.
 
Reduced row echelon forms should be unique, though.
 

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