# 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.

Given a topological space $X$ we can construct its suspension $\Sigma X$. This construction can be lifted to a functor from topological spaces to topological spaces that acts on arrows by sending a continuous map $f : X \to Y$ to its suspension $\Sigma f : \Sigma X \to \Sigma Y$ which is just $f$ in each “level” of the suspensions.