##### 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*}Find the eigenvalues of \(A\text{.}\)

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

Find the steady-state vectors of \(A\text{.}\)

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