Matrix Multiplication is not Commutative/Order 2 Square Matrices

Theorem
The operation of (conventional) matrix multiplication is not in general commutative.

Proof
Let $\mathbf A = \left[{a_{ij}}\right], \mathbf B = \left[{a_{jk}}\right]$ where $i \ne k$.

Then $\mathbf A \mathbf B$ is defined, but $\mathbf B \mathbf A$ is not.

Hence (trivially) the result.

Consider where $i = j = k$ and so $\mathbf A = \left[{a_i}\right], \mathbf B = \left[{a_i}\right]$ are square matrices.

Take for example where $i = 2$.

Then:

and:

and it can be seen that, in general, $\mathbf A \mathbf B \ne \mathbf B \mathbf A$.