Homotopic maps in the general linear group

Prove that the maps \mathrm{GL}(n,\mathbb R)\times \mathrm{GL}(n,\mathbb R)\longrightarrow \mathrm{GL}(2n,\mathbb R) that send (A,B) to the block matrices \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix} and \begin{pmatrix} AB & 0\\ 0 & 1\end{pmatrix} are homotopic.

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Homotopic maps in the general linear group