is $GL(n,\mathbb R)$ dense in $M(n,\mathbb R)$ Is $GL(n,\mathbb R)$ dense in $M(n,\mathbb R)$? I have proved it to be open,not closed,not connected but not sure about this property .How to do this?

Yes, $GL(n,\mathbb R)$ is dense in $M(n,\mathbb R)$. 
Using the determinant function $\det:M(n,\mathbb R)\rightarrow \mathbb R$, check that for small values of $\varepsilon$, the matrix $A+\varepsilon I_n$ is invertible.
