Duality with a stochastic matrix If I have a stochastic matrix $X$- the sum of each row is equal to $1$ and all elements are non-negative.
Given this property, how can I show that:
$x'X=x'$ ,  $x\geq 0$ 
Has a non-zero solution?
I'm assuming this has something to do with proving a feasible dual, but not may be wrong..

If your matrix $X$ is $n\times n$, then $X$ is the transition matrix of a Markov chain with state space $\lbrace  1 , 2 , \dots , n \rbrace$. The vector $x$ that you are looking for is a stationary measure for the Markov chain. 
The existence of such a non-zero $x$ can be proven using probability theory, but a purely linear approach falls under Perron-Frobenius theory. 
For instance, the averages ${1\over n}\sum_{k=1}^n X^k$ of the powers of $X$ converge to a matrix $M$ (this statement needs proof!). Then, it is easy to see that any row of $M$ can serve as $x'$ and solves $x'X=x'$. Note that the row sums of $M$ all equal 1, so $x\neq 0$.
As you suggest, maybe there is also an approach via linear programming. 
