Properties Of Matrix Multiplication

1. Not commutative. $A\times B \neq B\times A$
2. Associative. $(A\times B)\times C = A\times (B\times C)$

e.g. For $A \times B$ where $A$ is $m\times n$ matrix and $B$ is $n\times m$ matrix,
$A\times B$ is an $m\times m$ matrix,
$B\times A$ is an $n\times n$ matrix.

Identity matrix
Denoted as $I$ or $I_{n\times n}$
e.g. $$\begin{bmatrix} 1 & 0 & 0 \newline 0 & 1 & 0 \newline 0 & 0 & 1 \newline \end{bmatrix}$$
For any matrix $A$, $A\times I=I\times A=A$