Determining Uniqueness of Reduced Echelon Form

[ELB27]

Homework Statement

Is the reduced echelon form of a matrix unique? Justify your conclusion.
Namely, suppose that by performing some row operations (not necessarily following any algorithm) we end up with a reduced echelon matrix. Do we always end up with the same matrix, or can we get different ones? Note that we are only allowed to perform row operations, the “column operations”’ are forbidden.
Hint: What happens if you start with an invertible matrix? Also, are the pivots always in the same columns, or can it be different depending on what row operations you perform? If you can tell what the pivot columns are without reverting to row operations, then the positions of pivot columns do not depend on them.

2. Relevant definitions
A reduced echelon matrix satisfies the following 4 properties:
1. All zero rows (i.e. the rows with all entries equal 0), if any, are below all non-zero entries.
2. For any non-zero row its leading entry (aka pivot) is strictly to the right of the leading entry (pivot) in the previous row.
3. All pivot entries are equal 1;
4. All entries above the pivots are 0. (Note, that all entries below the pivots are also 0 because of the echelon form.)

The Attempt at a Solution

If I start with an invertible matrix, the reduced echelon form is always identity which is unique. If I start with only left-invertible matrix, every column will have a pivot (so I can tell before applying any row operations what the pivot columns are - all of them) and the reduced echelon form will be identity with possibly zero rows below it. If I start with a right invertible matrix, every row will have a pivot. Here I'm not sure that I can predict what the pivot columns are without any row operations to see which entries cancel (and this seems to depend on the algorithm used). I'm not sure how to approach the general case - it all seems very hand-wavy for me right now.

Any help/comments/suggestions on how to approach this are very welcome!

 

[ELB27]

Well, I checked in Wikipedia that says that the reduced echelo form is indeed unique but does not provide a proof. I did find a proof via induction here but somehow it went above my head (I'm not very familiar with induction). Other than that, it seems like people generally assume it to be unique. I would like to prove it by myself though, that's why I asked. The problem maker seems to imply some method for proving it via his hint, I just can't figure out what it is.