Let $X_n$ a markov chain with transition matrix given by $$\begin{bmatrix}0.7&0.3&0\\0.2&0.7&0.1\\0.4&0.1&0.5\end{bmatrix}$$
i) Find the stationary probability
ii)Starting from state "2", which is the expected number of steps that the process must do to return to the same state?
iii)This Markov Chain is reversible?
What I did
i)$$\pi P=\pi$$ $$0.7\pi_0+0.2\pi_1+0.4\pi_2=\pi_0$$ $$0.3\pi_0+0.7\pi_1+0.1\pi_2=\pi_1$$ $$\pi_0+\pi_1+\pi_2=1$$ solving this I get $$\begin{bmatrix}\pi_0 &\pi_1 &\pi_2\end{bmatrix}=\begin{bmatrix}0.09375& 0.4685 &0.4375\end{bmatrix}$$
ii)Let $T$ be the number of steps required to get back to 2 when start in 2, then $$E[T]=\frac{1}{\pi_2}=\frac{1}{0.4375}\approx 2.28$$
iii)How do I check that the chain is reversible? I need to check this $$\pi(x)P(x,y)=\pi(y)P(y,x)\forall x,y$$
EDIT: Following a tip from @John it is easy to see that this chain is not reversible, but supposing that $$P(x,y)>0\forall x,y$$ It would be very laborious check many cases to find one that is not valid.
Is there any other criteria to check reversibility?
EDIT: Check the balance condition and reversibility is the same thing?