Find the general solution of the system $X'=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}X$

Question

Find the general solution of $X'=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}X$

What I did: I calculated that the matrix has repeated eigenvalues of 0. When I plugged this in, I got that an eigenvector is $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. So one solution of the system is $c_1\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ for a constant $c_1$. But I can't find another solution.

Answer 1

This one is easier to solve directly, since the system reduces to $x' = y$ and $y' = 0$. I would say eigenvalues and so on would be overkill here.