Exercise 11

The proof of Lemma 1.5 contains a reference to the $i\neq j$ condition on the row combination operation.

• Write down a $2\!\times\!2$ matrix with nonzero entries, and show that the $-1\cdot\rho_{1}+\rho_{1}$ operation is not reversed by $1\cdot\rho_{1}+\rho_{1}$.

• Expand the proof of that lemma to make explicit exactly where it uses the $i\neq j$ condition on combining.