# 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.