<s>[INST] 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.
 [/INST] You want an $x\in\mathrm{Null}(A-I)$. Note that each column of the matrix $A-I$ adds up to $0$. 
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