Determinant function/Behavior under row operations/Fact

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

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

be a determinant function.

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

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

Proof, Write another proof