Determinant function/Behavior under row operations/Fact/Proof

Proof

(1) and (2) follow directly from multilinearity.
(3) follows from fact.
To prove (4), we consider the situation where we add to the ${}s$-th row the ${}a$-multiple of the ${}r$-th row, ${}r<s$. Due to the parts already proven, we have

${}\triangle {\begin{pmatrix}\vdots \\v_{r}\\\vdots \\v_{s}+av_{r}\\\vdots \end{pmatrix}}=\triangle {\begin{pmatrix}\vdots \\v_{r}\\\vdots \\v_{s}\\\vdots \end{pmatrix}}+\triangle {\begin{pmatrix}\vdots \\v_{r}\\\vdots \\av_{r}\\\vdots \end{pmatrix}}=\triangle {\begin{pmatrix}\vdots \\v_{r}\\\vdots \\v_{s}\\\vdots \end{pmatrix}}+a\triangle {\begin{pmatrix}\vdots \\v_{r}\\\vdots \\v_{r}\\\vdots \end{pmatrix}}=\triangle {\begin{pmatrix}\vdots \\v_{r}\\\vdots \\v_{s}\\\vdots \end{pmatrix}}\,.$

(5). If a diagonal element is ${}0$, then set ${}r=\max {\left\{i\mid a_{ii}=0\right\}}$. We can add to the ${}r$-th row suitable multiples of the ${}i$-th rows, ${}i>r$, in order to achieve that the new ${}r$-th row is a zero row, without changing the value of the determinant function. Due to (2), this value is ${}0$.

In case no diagonal element is ${}0$, we may obtain, by several scalings, that all diagonal element are ${}1$. By adding rows, we obtain furthermore the identity matrix. Therefore,

${}\triangle (M)=a_{11}a_{22}\cdots a_{nn}\triangle (E_{n})=a_{11}a_{22}\cdots a_{nn}\,.$

To fact