We present an interesting new procedure for computing the mean first passage times #opMFPTs#cp in an irreducible,N#pl1 state Markov chain. To compute the MFPTs to a given state we embed the submatrix of transition probabilities for theNremaining states in an augmented matrix. We perform successive repetitions of matrix reduction to reduce the augmented matrix to a column vector ofNelements which represent the MFPTs. The new procedure has a higher operation count than Gaussian elimination. We use matrix reduction to solve a numerical example problem for the MFPTs in a four‐state Markov chain.