is there a way to construct a transition matrix of a time-homogeneous, finite Markov chain such that the stationary distribution always has full support (this is equivalent to all states of the chain being positive recurrent).


I am looking for something quick that doesn't involve drawing the state-transition graph and checking for absorbing states etc.


Best would be something on the level of "check that every column sums to one". Is this just wishful thinking?



Constructing a transition matrix of a time-homogeneous, finite Markov chain with full support stationary distribution
When you write "**the** stationary distribution, I suppose you mean that the Markov chain is irreducible.
An easy sufficient condition is that all entries are strictly positive (and that the sum of each row is $1$ so it is a transition matrix). 


