Matrix determinant is a homomorphism

d e t : G L (𝑛, 𝕂) \to 𝕂 \times

is a group epimorphism, #m/thm/group where the codomain is the multiplicative group of the underlying field

Proof

From properties of the Matrix determinant, for any $𝐴, 𝐵 \in G L (𝑛, 𝕂)$ :

d e t (𝐴 𝐵) = d e t (𝐴) d e t (𝐵)

Hence $d e t$ is a homomorphism. For any $𝜆 \in 𝕂$ , there exists $𝐶 = 𝜆 𝟙$ with $d e t (𝐶) = 𝜆 d e t 𝟙 = 𝜆$ . Hence $d e t$ is an epimorphism.

Properties

#state/tidy | #lang/en | #SemBr