Adapt the method of row reduction to prove that the transpositions generate the symmetric group $S_n$.
I can prove the theorem that transpositions can generate $S_n$ but Artin asks to adapt row reduction. I don't know what he means. Kindly help to understand this question.
Every permutation of $S_n$ can be written as a permutation matrix in the group $GL_n(K)$, see here. For these matrices, do row reduction.