Given the below Markov transition matrix, calculate $p_{0,1}^9$
\begin{matrix} 0.5 & 0.5 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{matrix}
Method one, take the matrix powered to the 9. Then take the 0,1 index from that matrix. Result :
\begin{matrix} 0.00195 & 0.666 & 0.332 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{matrix}
Second method:https://en.wikipedia.org/wiki/Markov_chain#Transient_evolution Result : $0.5^8*0.5^1 = 0.00195$ (which weirdly coincides with index 0,0 of the powered matrix.
Am I overlooking something ?