$S:=\{A \in M(3,\mathbb R) : A^TA(e_1)=e_1 ; A^TA(e_2)=A^TA(e_3)=0\}$ ; then does $S$ contain a nilpotent matrix?

Question

$S:=\{A \in M(3,\mathbb R) : A^TA(e_1)=e_1 ; A^TA(e_2)=A^TA(e_3)=0\}$ , where $e_1,e_2,e_3$ are the standard vectors of $\mathbb R^3$ . Then is it true that $S$ contains a nilpotent matrix ? Does $S$ contain a matrix of rank $1$ ?

Answer 1

Consider mapping $e_1$ to $e_2$ and doing something trivial with the other two basis vectors. This answers both parts if you squint long enough and think about the definition for a rank $1$ matrix.

Answer 2

Assume $A$ were nilpotent, say $A^n=0$. Then we should have $(A^T)^nA^ne_1=0$.

The projection onto $e_1\Bbb R$ clearly is of rank $1$ and $\in S$.