Let $n\in\mathbb N$ and let $\mathbb F\in\{\mathbb R,\mathbb C\}$. Regard $GL(n,\mathbb F)$ as the open smooth submanifold of $M(n,\mathbb F)$, and in the case $\mathbb F=\mathbb C$ regard $M(n,\mathbb C)$ as a real [vector space](/page/Vector%20Space) of dimension $2n^2$. Then the multiplication map
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\begin{align*}
m:GL(n,\mathbb F)\times GL(n,\mathbb F)\to GL(n,\mathbb F),\qquad (A,B)\mapsto AB
\end{align*}