###### Exercise 4

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 \(A\text{?}\)