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.

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