Let $A=\operatorname{diag}(2,3,4)$ and $B=\begin{bmatrix}1&2&0\\0&1&1\\1&0&1\end{bmatrix}$. Then $AB=\begin{bmatrix}2&4&0\\0&3&3\\4&0&4\end{bmatrix}$. Left multiplication by a diagonal matrix scales rows. In contrast, $BA$ scales the columns of $B$.
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