Proof of Theorem DEM

Swapping rows $i$ and $j$ of the identity matrix will create $\elemswap{i}{j}$ (Definition ELEM), so \begin{align*} \detname{\elemswap{i}{j}} &=-\detname{I_n}&&\knowl{./knowls/theorem.DRCS.knowl}{\text{Theorem DRCS}}\\ &=-1&&\knowl{./knowls/theorem.DIM.knowl}{\text{Theorem DIM}}\\ \end{align*}

Multiplying row $i$ of the identity matrix by $\alpha$ will create $\elemmult{\alpha}{i}$ (Definition ELEM), so \begin{align*} \detname{\elemmult{\alpha}{i}} &=\alpha\detname{I_n}&&\knowl{./knowls/theorem.DRCM.knowl}{\text{Theorem DRCM}}\\ &=\alpha(1)=\alpha&&\knowl{./knowls/theorem.DIM.knowl}{\text{Theorem DIM}}\\ \end{align*}

Multiplying row $i$ of the identity matrix by $\alpha$ and adding to row $j$ will create $\elemadd{\alpha}{i}{j}$ (Definition ELEM), so \begin{align*} \detname{\elemadd{\alpha}{i}{j}} &=\detname{I_n}&&\knowl{./knowls/theorem.DRCMA.knowl}{\text{Theorem DRCMA}}\\ &=1&&\knowl{./knowls/theorem.DIM.knowl}{\text{Theorem DIM}}\\ \end{align*}