Group homomorphism/Determinant/Example

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

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

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