Proof of Theorem EMN

We show that we can row-reduce each elementary matrix to the identity matrix. Given an elementary matrix of the form $\elemswap{i}{j}$, perform the row operation that swaps row $j$ with row $i$. Given an elementary matrix of the form $\elemmult{\alpha}{i}$, with $\alpha\neq 0$, perform the row operation that multiplies row $i$ by $1/\alpha$. Given an elementary matrix of the form $\elemadd{\alpha}{i}{j}$, with $\alpha\neq 0$, perform the row operation that multiplies row $i$ by $-\alpha$ and adds it to row $j$. In each case, the result of the single row operation is the identity matrix. So each elementary matrix is row-equivalent to the identity matrix, and by Theorem NMRRI is nonsingular.