On the Levy-Desplanques theorem proof: http://planetmath.org/levydesplanquestheorem, they only prove the second inequality for M = i. What about if i ≠ M? e.g. if we are doing it for the first line on a singular matriz and M ≠ 1 we can't get to the second inequality. I thought that to prove: A strictly diagonally dominant matrix is non-singular (1) You had to prove: A singular matrix is not strictly diagonally dominant (2). Howver, they only prove (2) for i = M, whereas it should be for all i! What am I missing here? I can't understand how proving for only i 0 M constitutes a proof, and I can't prove it for all i.