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:


 * $\mathbf A \mathbf B = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$

which works out as:
 * $\begin{bmatrix} a_{11} b_{11} + a_{12} b_{21} & a_{11} b_{12} + a_{12} b_{22} \\ a_{21} b_{11} + a_{22} b_{21} & a_{21} b_{12} + a_{22} b_{22} \end{bmatrix}$

and:
 * $\mathbf B \mathbf A = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

which works out as:
 * $\begin{bmatrix} b_{11} a_{11} + b_{12} a_{21} & b_{11} a_{12} + b_{12} a_{22} \\ b_{21} a_{11} + b_{22} a_{21} & b_{21} a_{12} + b_{22} a_{22} \end{bmatrix}$

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