Let MT be the mean 3rst passage matrix for an n-state ergodic Markov chain with a transition matrix T . We partition T as a 2× 2 block matrix and show how to reconstruct MT e@ciently by using the blocks of T and the mean 3rst passage matrices associated with the non-overlapping Perron complements of T . We present a schematic diagram showing how this method for computing MT can be implemented in parallel. We analyse the asymptotic number of multiplication operations necessary to compute MT by our method and show that, for large size problems, the number of multiplications is reduced by about 1=8, even if the algorithm is implemented in serial. We present 3ve examples of moderate sizes (of orders 20–200) and give the reduction in the total number of Cops (as opposed to multiplications) in the computation of MT . The examples show that when the diagonal blocks in the partitioning of T are of equal size, the reduction in the number of Cops can be much better than 1=8. Copyright ? 2001 John Wiley & Sons, Ltd.