Find vector $x \not= 0$ that satisfies the equation $Ax = x$.

Question

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.

Answer 1

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

$$B=A-I= \begin{bmatrix} -1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{3} & -1 & 1 & 0 & 0 \\ \frac{1}{3} & 0 & -1 & 0 & \frac{1}{2} \\ \frac{1}{3} & \frac{1}{2} & 0 & -1 & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} & -1 \\ \end{bmatrix} $$

Answer 2

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