Determinant function/Behavior under row operations/Fact

Let ${\displaystyle {}K}$ be a field and ${\displaystyle {}n\in \mathbb {N} _{+}}$. Let

${\displaystyle \triangle \colon \operatorname {Mat} _{n}(K)\cong {\left(K^{n}\right)}^{n}\longrightarrow K}$

be a determinant function.

Then ${\displaystyle {}\triangle }$ fulfills the following properties.

1. If a row of ${\displaystyle {}M}$ is multiplied with ${\displaystyle {}s\in K}$, then ${\displaystyle {}\triangle }$ is multiplied by ${\displaystyle {}s}$.
2. If ${\displaystyle {}M}$ contains a zero row, then ${\displaystyle {}\triangle (M)=0}$.
3. If in ${\displaystyle {}M}$ two rows are swapped, then ${\displaystyle {}\triangle }$ is multiplied with the factor ${\displaystyle {}-1}$.
4. If a multiple of a row is added to another row, then ${\displaystyle {}\triangle }$ does not change.
5. If ${\displaystyle {}\triangle (E_{n})=1}$, then, for an upper triangular matrix, we have ${\displaystyle {}\triangle (M)=a_{11}a_{22}\cdots a_{nn}}$.