Connected spaces of $M(n,\mathbb R)$

Question

Consider $M(n,\mathbb R)$, the space of all $ n\times n$ matrices over $\mathbb R$.Which of the following are connected? a.$O(n)$ the set of all orthogonal matrices

b.$GL(n,\mathbb R)$ set of all non-singular matrices over $\mathbb R$

c.$SL(n,\mathbb R)$ set of all non-singular matrices over $\mathbb R$ with determinant equals one.

d.set of all nilpotent matrices.

Answer:since the determinant map is continuous and continuous image of a connected set is connceted using them I have concluded (a),(b) is not connected since image set of (a) is $\{-1,1\}$and of (b) is $(-\infty,0) \cup (0,\infty)$. Is it correct?But how to approach the others

Answer 1

Hint: for (c), you can prove the result by showing that any matrix of $SL_n$ is a product of matrices of the form $I_n+\lambda E_{ij}$ with $i\neq j$, and create a path to the identity by shrinking the $\lambda$ to $0$.

Hint: for (d), if $n$ is nilpotent, then so is $tn$, for $t\in[0,1]$: this is a continuous path of nilpotent matrices that stops at $0$, thus...