General Linear Group to Determinant is Homomorphism

Theorem
Let $\GL {n, \R}$ be the general linear group over the field of real numbers.

Let $\struct {\R_{\ne 0}, \times}$ denote the multiplicative group of real numbers.

Let $\det: \GL {n, \R} \to \struct {\R_{\ne 0}, \times}$ be the mapping:
 * $\mathbf A \mapsto \map \det {\mathbf A}$

where $\map \det {\mathbf A}$ is the determinant of $\mathbf A$.

Then $\det$ is a group homomorphism.

Its kernel is the special linear group $\SL {n, \R}$.

Proof
From Determinant of Matrix Product:
 * $\map \det {\mathbf A \mathbf B} = \map \det {\mathbf A} \, \map \det {\mathbf B}$

which is seen to be a group homomorphism by definition.

The special linear group $\SL {n, \R}$ is the subset of $\GL {n, \R}$ such that:
 * $\forall \mathbf A \in \SL {n, \R}: \map \det {\mathbf A} = 1$

From Real Multiplication Identity is One:
 * $1$ is the identity of the multiplicative group of real numbers.

It follows by definition that $\SL {n, \R}$ is the kernel of the $\det$ mapping.