# Where is the gap in the proof of uniqueness of the row reduced echelon form?

The author of the book I'm reading says that the following proof has a gap,

but I don't know where it is.

Here is the proof which the author claims that it has a gap.

First, The author defines that 'row-equivalentness',

$$A$$ and $$B$$ are row-equivalent if there exist elementary matrices $$E_i$$ such that

$$E_1 E_2 \cdots E_k A = B.$$

where elementary matrix is the matrix corresponding to the elementary row operations. (exchange rows, multiply a row by given scalar, add $$i$$th row by $$c$$ multiple of $$j$$th row)

And shows that row-equivalentness is an equivalence relation.

So if $$A$$ had $$R_1, R_2$$ as its row reduced echelon form,

then $$R_1 \equiv R_2$$.

But the only way in which apply elementary row operations to row reduced echelon form to make

again row reduced echelon form is to exchange zero rows.

Therefore $$R_1 = R_2$$, i.e., its row reduced echelon form is unique.

Where is the gap? I can't see where it is.

• Multiplying a zero row by a constant, or adding a multiple of a zero row to another row will also leave the RREF unchanged. Not sure whether that's what the author means. Mar 21, 2023 at 2:44