Outer products are useful nonsquare products: $\begin{bmatrix}1\\2\end{bmatrix}\begin{bmatrix}3&4&5\end{bmatrix}=\begin{bmatrix}3&4&5\\6&8&10\end{bmatrix}$. The result has rank one because every column is a multiple of $(1,2)^T$. Reversing the factors in this example is undefined.
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