Consider the stochastic matrix $P=\left[\begin{array}{11}0 & 1 1 1 & O\end{array} ight]$. (a) Find a formula for $P^{k}$. (b) Explain why no power of $P$ has all positive entries. Thus $P$ is not regular. (c) Find the steady-state vector $\vec{q}$ for $P$. Check that $P Wec{q}=\vec{q}$. (d) Invent an initial state vector $\overrightarrow{x_{0}}$. such that the Markov chain $\overrightarrow{x_{k}}=P^{k} \overrightarrow{x_{0}}$. does not converge to $\vec{q} $ as $k ightarrow \infty$. Explain why convergence fails. SP.JG. 343