General Linear Group is Group

Theorem
Let $$K$$ be a field.

The set of all invertible order-$$n$$ square matrices over $$K$$ is a group under matrix multiplication.

This group is called the General Linear Group and is denoted $$GL \left({n, K}\right)$$

The field itself is usually $$\mathbb{R}$$, $$\mathbb{Q}$$ or $$\mathbb{C}$$, but can be any field.

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 $$GL \left({n, K}\right)$$ is closed.

G1: Associativity
Matrix Multiplication is Associative.

G2: Identity
The Identity Matrix serves as the identity of $$GL \left({n, K}\right)$$.

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