Using the given transition matrix and assuming the initial probability vector is [1 0 0], find the probability vectors for the next four steps of the Markov chain. (This initial probability vector indicates that all the members are Democrats.)
In a certain city the Democratic, Republican, and Consumer parties have members of their parties on the city council. The probability of a member of this party winning in any election depends on the proportional membership of his or her party at the time of the election. The probabilities for all these parties winning are given by the following transition matrix P.

Party in office next term Party in office now Democrat D R 0.5 0.4 0.5 0.1 0.4 0.5 0.1 0.3 0.3 0.4 Republican Consumer 0.4