Let P = \left [\begin{matrix} .5 & .2 & .3 \\ .3 & .8 & .3 \\ .2 & 0 & .4 \end{matrix} \right ] and x_{0} = \left [ \begin{matrix} 1 \\ 0 \\ 0 \end{matrix} \right ] . Consider a system whose state is described by the Markov chain x_{k+1} = Px_{k} , \text{for} k = 0, 1, . . . What happens to the system as time passes? Compute the state vectors x_{1} , . . . , x_{15} to find out.