##### Exercise4

Consider the stochastic matrix

\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 0.2 \amp 0.2 \\ 0 \amp 0.6 \amp 0.2 \\ 0 \amp 0.2 \amp 0.6 \\ \end{array}\right] \text{.} \end{equation*}
1. Find the eigenvalues of $$A\text{.}$$

2. Do the conditions of the Perron-Frobenius theorem apply to this matrix?

3. Find the steady-state vectors of $$A\text{.}$$

4. What can we guarantee about the long-term behavior of a Markov chain defined by the matrix $$A\text{?}$$

in-context