I needed to find the limiting distribution of the matrix 

> 
> $$\pmatrix{ 0 & 0 & 1 \\ 1 & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & 0}$$
> 
> 
> 

Instead of $\pi$ I'll use $A, B$ and $C$ because that might be easier to read.

To solve this, I got

$$A = B + \frac{1}{2}C$$
$$B = \frac{1}{2}C$$
$$C = A$$

Now to solve this, I ended up getting $A = C$ and I got $A = 2B$. I also need to satisfy $A + B + C = 1$. So, I said let $A = \frac{1}{4}, B = \frac{1}{2}, C = \frac{1}{4}$, which satisfies all my equations. But in the answers they say that $A = \frac{1}{5}, B = \frac{2}{5}, C = \frac{1}{5}$, which also satisfy the equations.

How did I get mine wrong? Obviously it matters what the limiting distribution is, so how am I supposed to tell which is the correct one?