Does there exist a choice of the parameter $c$ such that the M.C. is time reversible? If so, for what values of $c$? If not, why? Consider a Markov chain with state space $S = ${$1,2$}. Its transition probabilities are
$P_{11} = 0, P_{12} = 1, P_{21} = 1−c, P_{22} = c.$
Does there exist a choice of the parameter $c$ such that the M.C. is time reversible? If so, for what
values of $c$? If not, why?
having trouble with with one, any ideas?

We have
$$
P = \begin{pmatrix} 0 & 1\\1-c&c\end{pmatrix}.
$$
If $c\ne0$, then 
$$\pi = \left(\frac{1-c}{2-c}, \frac1{2-c}\right) $$
is a stationary distribution for $P$ (and is unique since $P$ is irreducible and the state space is finite). Then the time reversal with respect to $\pi$ would be
$$P^*_{ij} = \frac{\pi(j)}{\pi(i)}P_{ji}. $$
If you compute $P^*$, you'll find that $P^*=P$, so the Markov chain is time reversible for all $0<c\leqslant1$.
