why can’t we subtract the third row from the second and subtract the first row from the third row to get to the row echelon form so the second row becomes 0,1,-1 instead of 0,1,0?
Any linear combination of rows is valid from a math p.o.v., but we are designing an algorithm here. So as I commented on your other thread asking essentially the same question, why make things more complicated? What is your decision process for picking the row to use? He proceeds “top down” here in a predictable manner.