IF we are given a markov chain probability matrix represented by
$$P = \left(\begin{matrix} a&&b&&c\\d&&e&&f\\g&&h&&i\end{matrix}\right)$$
and $a+b+c=d+e+f=g+h+i=1$ with an initial state $(x, y, z)$
Is there a way to know whether after how many iterations the matrix becomes independent of initial state? I know that in excel sheet, we can continue multiplying P a number of times until we get some thing like say
$$P=\left(\begin{matrix} a&&b&&c\\a&&b&&c\\a&&b&&c\end{matrix}\right)$$
but is it possible to find n at which it achieves this state without using an excel sheet ?