1 0 0 0
c1 0 + c2 1 + c3 0 = 0 so c1 = 0, c2=0 and c3=o----->independent
0 0 1 0
I think you are trying to write
and derive from that that those vectors are independent. But that is the identity matrix, not a general "row echelon matrix". A three by three row echelon matrix would be of the form:
so you should be looking at
Also, I see nothing in your question that restricts this to 3 by 3 matrices. You should be able to do a general proof.