  Questions are posed regarding the influence that the column sums of the
transition probabilities of a stochastic matrix (with row sums all one) have on
the stationary distribution, the mean first passage times and the Kemeny
constant of the associated irreducible discrete time Markov chain. Some new
relationships, including some inequalities, and partial answers to the
questions, are given using a special generalized matrix inverse that has not
previously been considered in the literature on Markov chains.
