Find vector $x \not= 0$ that satisfies the equation $Ax = x$. Given a matrix A:
$$A = \begin{bmatrix}
        0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \\
        \frac{1}{3} & 0 & 1 & 0 & 0 \\
        \frac{1}{3} & 0 & 0 & 0 & \frac{1}{2} \\
        \frac{1}{3} & \frac{1}{2} & 0 & 0 & 0 \\
        0 & \frac{1}{2} & 0 & \frac{1}{2} & 0 \\
    \end{bmatrix}$$
I have to find a vector $x \not= 0$ that satisfies the equation $Ax = x$.
I have no idea how to approach this. Any hints would be appreciated.
Thanks in advance.

HINT
Note that
$$Ax = x\iff Ax-Ix=0\iff (A-I)x=0$$
thus the original problem is equivalent to find the null space for 
$$B=A-I= \begin{bmatrix}
        -1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \\
        \frac{1}{3} & -1 & 1 & 0 & 0 \\
        \frac{1}{3} & 0 & -1 & 0 & \frac{1}{2} \\
        \frac{1}{3} & \frac{1}{2} & 0 & -1 & 0 \\
        0 & \frac{1}{2} & 0 & \frac{1}{2} & -1 \\
    \end{bmatrix}
$$
