I have this system of five equations
$$Av_1=-e_1,\quad Ae_5=-e_1,\quad A(e_2-e_3)=e_3-e_2,\\ (A-2I)^2(e_2+e_3)=0,\quad (A-2I)^2(e_1+e_3-e_4)=0$$
for some $A\in\Bbb R^{5\times 5}$, where $e_1,e_2,\ldots,e_5$ is the standard basis of $\Bbb R^5$. I dont know if the system is solvable or how to approach it solution. That is: I want to know if there is a way to find a explicit definition for $A$.
Note that to define explicitly such $A$ I just need to know what are the $v_k$ of $Ae_k=v_k$ for all $k=1,\ldots,5$.
I would like to know also some book or other reference where I can learn how to approach these kind of problems.