Determinant/Recursively/Alternating/Swap property/Fact

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

$\operatorname {Mat} _{n}(K)=(K^{n})^{n}\longrightarrow K,M\longmapsto \det M,$

has the following properties.

1. If in ${}M$ two rows are identical, then ${}\det M=0$. This means that the determinant is alternating.
2. If we exchange two rows in ${}M$, then the determinant changes with factor ${}-1$.

Write a proof