Group homomorphism/Determinant/Example

For a field ${}K$ and ${}n\in \mathbb {N} _{+}$, the determinant

$\det \colon \operatorname {GL} _{n}\!{\left(K\right)}\longrightarrow K^{\times },M\longmapsto \det M,$

is a group homomorphism. This follows from the multiplication theorem for the determinant and fact.