I need to perform multi-step ahead prediction using multivariate Markov model. Do we need to update the transition matrix after each prediction or use the same. How can we update it based on prediction. If I don't update it and just update the state matrix, it is predicting same state always even for next 60 steps.
It depends, if you think the transition probabilities are time-varying, then you should update it. One way to update the probabilities by using Bayesian updating, Kim and Nelson have a book "State space models with regime switching" with online examples found here (in Gauss, but you can probably translate it to code suiting your software). I think chapter 9 covers the relevant part, but I will give you an intuitive explanation.
We start by assuming a prior distribution, the Beta distribution if there are only two possible states and the Dirichlet distribution if there are multiple states (Dirichlet is the multivariate Beta distribution). It is up to you to set the parameters of the prior distribution, common is to use 1 for all parameters. This specification means that all states are equally likely (at the beginning). Update the parameters by counting the number of transitions from state i to j (say x) the corresponding updated parameter will then be 1 + x. Then you need to sample a value for the transition probability using (e.g.) the Gibbs sampler. The new transition probability will then be higher/lower if we have counted many/few transitions from state i to j.
There are many more (infinitely?) ways to find new transition probabilities (e.g. making a model just to model and predict these probabilities). But I would advise that you use a model you are familiar with and not learn a totally new method for a relatively small part of your research.
ps It is probably logical that you get the same value after 60 steps. Suppose that the transition probability from state i to 1 is 0.9 (for every i), then of course we expect that you will always be in state 1. You should check your transition matrix and check which states you pass if you always choose the 'highest probability path'.