General Linear Group is Group

Theorem
Let $K$ be a field.

Let $\GL {n, K}$ be the general linear group of order $n$ over $K$.

Then $\GL {n, K}$ is a group.

Proof
Taking the group axioms in turn:

G0: Closure
The matrix product of two $n \times n$ matrices is another $n \times n$ matrix.

The matrix product of two invertible matrices is another invertible matrix.

Thus $\GL {n, K}$ is closed.

G1: Associativity
Matrix Multiplication is Associative.

G2: Identity
From Unit Matrix is Unity of Ring of Square Matrices, the unit matrix serves as the identity of $\GL {n, K}$.

G3: Inverses
From the definition of invertible matrix, the inverse of any invertible matrix $\mathbf A$ is $\mathbf A^{-1}$.