Linear and multilinear algebra; matrix theory Example

Determinant as a group homomorphism

Builds on Group homomorphism Kernel of a homomorphism

Example

Let $n$ be a positive integer and let $\mathrm{GL}_n(\mathbb{R})$ be the group of invertible $n \times n$ real matrices under multiplication. The determinant is a map $\det \colon \mathrm{GL}_n(\mathbb{R}) \to \mathbb{R}^\times$ to the group of nonzero real numbers under multiplication. It is a group homomorphism, because $\det(AB) = \det(A) \det(B)$ for all $A, B$ .

Remarks

The kernel of $\det$ is the special linear group $\mathrm{SL}_n(\mathbb{R})$ of matrices of determinant $1$ , so this group is normal in $\mathrm{GL}_n(\mathbb{R})$ . The determinant is surjective onto $\mathbb{R}^\times$ , so $\mathrm{GL}_n(\mathbb{R}) / \mathrm{SL}_n(\mathbb{R}) \cong \mathbb{R}^\times$ . The same holds over each field $F$ , with $\mathrm{GL}_n(F)$ , $\mathrm{SL}_n(F)$ and $F^\times$ .

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Group concept
Group homomorphism definition
Kernel of a homomorphism definition
Determinant as a group homomorphism example

Tag	`0YZ3T5Q7XN`
URI	`m2://0YZ3T5Q7XN`
Kind	example
Writing	W0 stub
MSC	15A15
Instance of	Group homomorphism
Uses	Group homomorphism, Kernel of a homomorphism

Determinant as a group homomorphism

Example

Remarks

Reading path

Further reading

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