Determinant/Recursively/Alternating/Fact

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

Then the

determinant

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

is

alternating.

Proof, Write another proof