EDIT: I decided to ask this question after working on this particular problem. I had the stupidity to think that row-reduced = row reduced echelon form. Brain fart, nothing more to see here...
Wikipedia lists the following two conditions for a matrix to be in reduced row echelon form, in agreement with my Linear Algebra book by Hoffman and Kunze:
- It is in row echelon form.
- Every leading coefficient is 1 and is the only nonzero entry in its column
I am told that the reduced row echelon form of a matrix is unique. However, according to the above two conditions only, then is it not possible to have multiple forms of RREF through row interchanges? Ergo, would not an alternative RREF for $I_3$ be the following?
\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}