Suppose you live in a country with three political parties \(P\text{,}\) \(Q\text{,}\) and \(R\text{.}\) We use \(P_k\text{,}\) \(Q_k\text{,}\) and \(R_k\) to denote the percentage of voters voting for that party in election \(k\text{.}\)

Voters will change parties from one election to the next as shown in the figure. We see that 60% of voters stay with the same party. However, 40% of those who vote for party \(P\) will vote for party \(Q\) in the next election.

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  1. Write expressions for \(P_{k+1}\text{,}\) \(Q_{k+1}\text{,}\) and \(R_{k+1}\) in terms of \(P_k\text{,}\) \(Q_k\text{,}\) and \(R_k\text{.}\)

  2. If we write \(\xvec_k = \threevec{P_k}{Q_k}{R_k}\text{,}\) find the matrix \(A\) such that \(\xvec_{k+1} = A\xvec_k\text{.}\)

  3. Explain why \(A\) is a stochastic matrix.

  4. Suppose that initially 40% of citizens vote for party \(P\text{,}\) 30% vote for party \(Q\text{,}\) and 30% vote for party \(R\text{.}\) Form the vector \(\xvec_0\) and explain why \(\xvec_0\) is a probability vector.

  5. Find \(\xvec_1\text{,}\) the percentages who vote for the three parties in the next election. Verify that \(\xvec_1\) is also a probabilty vector and explain why \(\xvec_k\) will be a probability vector for every \(k\text{.}\)

  6. Find the eigenvalues of the matrix \(A\) and explain why \(E_1\) is a one-dimensional subspace of \(\real^3\text{.}\) Then verify that \(\vvec=\threevec{1}{2}{2}\) is a basis vector for \(E_1\text{.}\)

  7. As every vector in \(E_1\) is a scalar multiple of \(\vvec\text{,}\) find a probability vector in \(E_1\) and explain why it is the only probability vector in \(E_1\text{.}\)

  8. Describe what happens to \(\xvec_k\) after a very long time.