TITLE: Is the set $ {\rm SL }(n, \mathbb {R}) $ connected in ${\rm M }(n, \mathbb {R}) $
QUESTION [13 upvotes]: Is the set $ {\rm SL }(n, \mathbb {R}) $ connected in ${\rm M }(n, \mathbb {R}) $?
Can you  give me hints? I have only concept on  following tags.

REPLY [4 votes]: Here is an alternative proof of connectedness if you don't mind using something less elementary.
Let $A=USV^T$ be a singular value decomposition over $\mathbb{R}$. As $\det A=1$, we may assume WLOG that $\det U=\det V=1$ and hence $U=e^{K_1}$ and $V=e^{K_2}$ for some skew symmetric matrices $K_1,K_2$. Let $S=\operatorname{diag}(\sigma_1,\sigma_2,\ldots,\sigma_n)$. Then $I$ is pathwise connected to $A$ via the path
$$f(t)=e^{tK_1}\operatorname{diag}(\sigma_1^t,\sigma_2^t,\ldots,\sigma_n^t)e^{-tK_2}.$$