# Constructing a transition matrix of a time-homogeneous, finite Markov chain with full support stationary distribution

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?

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).