Subspace of $M(3,\Bbb R)$ true or false questions

Question

Let $W \subset M(3,\Bbb R)$ be a vector subspace such that $\dim W=7$ I need to determine if this statements are true of false:

1. $\exists\: A \in W$ such that $A$ is not null and A is diagonalizable over $\Bbb R$
2. $\exists\: A \in W$ such that $rk(A) = 2$
3. $\exists\: A \in W$ such that $A$ is not diagonalizable over $\Bbb C$
4. $\exists\: A \in W$ such that $A$ is not nilpotent

I'm completely stuck on this problem. I tried to write $A$ in terms of a base of $W$ but I can't go further in any of this points. For the third point I believe I need to demonstrate if exist a nilpotent matrix that is not null but again, I really haven't any ideas. Any hint or even solution would be much appreciated. Thank you in advance!