Markov chain - Regular transition matrix

Question

I have to prove that this transition matrix is regular but how can I prove it without having to multiply it n times?

I know when you raise it to the 2nd power you have to check if the zeros in the first matrix are at the same entries as in the second one but anyone has a more in depth explanation?

Answer 1

A stochastic matrix is regular if it's irreducible and has at least one non-zero entry on its main diagonal. It's easy to show that that your matrix is irreducible, since every state communicates with state $1$, and state $\ i\ $ communicates with state $\ i+1\ $ for $\ i=1,2,3,4\ $, and the first entry on its main diagonal is non-zero. Therefore it's regular.