$GL_n(\mathbb{R})$ is the union of two path connected subsets (Michael Artin's Algebra). I have that $SL_n(\mathbb{R})$ is path connected and generated by elementary matrices of the form $I + a e_{i,j} (i \neq j)$, where $e_{i,j}$ is the matrix with $1$ at position $(i,j)$ and zero elsewhere.  I also have that $GL_n(\mathbb{R})$ is path connected and generated by elementary matrices of the above type together with ones of the form $I + c e_{i,i}$.  So clearly I already have a union of two path connected subsets, but I was wondering if there is a smaller second set, for instance, matrices that have at least one factor of the second type.  Hints are welcome.

At first i wanted to mention that $GL_n(\mathbb{R})$ is not connected, the determinant is continuous and the image of a connected space under a continuous function is a connected space . 
But the image is $\mathbb{R}\setminus \{0\}$ which is surely not connected..
On the other hand 
$GL_n(\mathbb{R})$ is surely
$$GL_n(\mathbb{R})=GL_n^- (\mathbb{R}) \cup  GL_n^+ (\mathbb{R})$$
Now think of Row reduced Echolon form and you will find continuous generators, for each of them.
For the generating system of $GL_n^+ (\mathbb{R})$ you make the following: 
\begin{align*}
B_{b}^\lambda=(b_{k,l})_{1\leq k,l\leq n}=
\left\{
\begin{array}{ll}
\lambda & k=i \text{ and } l=j\\
0 & \text{ else }\\
\end{array}
\right.
\end{align*}
\begin{align*}
\gamma_{b}(t)=E_n+t\cdot B_{b}^\lambda
\end{align*}
\begin{align*}
C(t)=(c_{k,l}(t))_{1\leq k,l \leq n}&=\left\{
\begin{array}{cl}
1 & k=l \text{ with } k,l\neq i,j,\\
-\sin{(\pi \cdot t)} & k=i,\ l=j\\
\sin{(\pi \cdot t)} & k=j, \ l=i\\
\cos{(\pi \cdot t)} & k=l=i \\
\cos{(\pi \cdot t)} & k=l=j \\
0 & \text{else}\\
\end{array}
\right.\\
\gamma_{c)}(t)&=C(t)
\end{align*}
\begin{align*}
A(t)=(a_{kl})_{1\leq k,l\leq n}=
\left\{
\begin{array}{ll}
1 & k=l\neq i\\
1+t\cdot (\lambda-1) & k=l=i\\
0 & \text{else}\\
\end{array}
\right.
\end{align*}
with $\lambda > -1$
$t$ is always in $[0,1]$.
