TITLE: Reduced Row Echelon form without scalar multiplication?
QUESTION [1 upvotes]: Is it possible to transform any matrix to row reduced echelon form without using the row operation that multiplies a row by a scalar?

REPLY [2 votes]: No. In the case of square matrices, all the other row operations either leave the determinant unchanged or negates it. So if the original matrix has determinant $\notin\{-1,0,1\}$, then its reduced row echelon form is the identity matrix (with determinant $1$) but cannot be reached using the other row operations.
{"set_name": "stack_exchange", "score": 1, "question_id": 874175}