General linear group

Matrix determinant is a homomorphism

The Matrix determinant when applied to the General linear group

det:GL(𝑛,𝕂)→𝕂×

is a group epimorphism, group where the codomain is the multiplicative group of the underlying field

Proof

From properties of the Matrix determinant, for any 𝐴,𝐵 ∈GL(𝑛,𝕂):

det(𝐴𝐵)=det(𝐴)det(𝐵)

Hence det is a homomorphism. For any 𝜆 ∈𝕂, there exists 𝐶 =𝜆𝟙 with det(𝐶) =𝜆det𝟙 =𝜆. Hence det is an epimorphism.

Properties

• ker⁡det =SL(n,𝕂) the Special linear group.

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