Matrix multiplication is generally not commutative. Let $A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$ and $B=\begin{bmatrix}1&0\\1&1\end{bmatrix}$. Then $AB=\begin{bmatrix}2&1\\1&1\end{bmatrix}$ while $BA=\begin{bmatrix}1&1\\1&2\end{bmatrix}$. Even the existence of $AB$ does not guarantee that $BA$ is defined.
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