In this chapter, we consider the mean first passage matrix associated with a Markov chain on n states having an irreducible transition matrix T . In section 6.1, we give a basic result showing how the mean first passage matrix can be computed from the group inverse of I−T, and discuss some of its consequences. A triangle inequality for the entries in the mean first passage matrix is derived in section 6.2; we then go on to extend that inequality to the entries in certain symmetric inverse M-matrices. Section 6.3 addresses the problem of determining which positive matrices are realisable as mean first passage matrices, and links that problem to a special case of the inverse M-matrix problem. In section 6.4, we consider a mean first passage matrix in partitioned form as a 2×2 block matrix, and show how those blocks can be computed from the mean first passage matrices corresponding to certain stochastic complements of the original transition matrix. Finally, in section 6.5, we discuss some of the properties of the Kemeny constant associated with an irreducible stochastic matrix; in particular, we find the minimum value of the Kemeny constant over all transition matrices whose directed graphs are subordinate to a given directed graph.