Diagonalisation and powers Problem set B

Write each matrix in the form \(PDP^{-1}\).

1. (a) \(\begin{pmatrix}6 & 1 \\ 2 & 5\end{pmatrix}\)
2. (b) \(\begin{pmatrix}1 & 3 \\ 2 & 2\end{pmatrix}\)

For \(M = \begin{pmatrix}6 & 1 \\ 2 & 5\end{pmatrix}\), find \(M^n\) in terms of \(n\), and hence find \(M^4\).

\(M = \begin{pmatrix}2 & 3 \\ 4 & 3\end{pmatrix}\) has eigenvalues 6 and −1. Diagonalise \(M\) and use this to find \(M^5\).

A matrix \(M\) has eigenvalue \(5\) with eigenvector \(\begin{pmatrix}1 \\ 3\end{pmatrix}\) and eigenvalue \(2\) with eigenvector \(\begin{pmatrix}1 \\ 2\end{pmatrix}\).

1. (a) Find \(M\).
2. (b) Find \(M^{3}\).

Each year 30% of the customers of one phone network switch to its rival, and 20% of the rival's customers switch to it. This is modelled by \(T = \begin{pmatrix}0.7 & 0.2 \\ 0.3 & 0.8\end{pmatrix}\). At the start there are 10 000 customers on each network. (a) Find the numbers after 3 years. (b) Find the matrix that \(T^n\) approaches, and the numbers in the long term.

Each year every adult seal produces 1.6 pups, 30% of the pups survive to become adults, and 80% of the adults survive. Let \(y_n\) and \(a_n\) be the numbers of young and adult seals in year \(n\).

1. (a) Write down the matrix \(M\) with \(\begin{pmatrix}y_{n+1} \\ a_{n+1}\end{pmatrix} = M\begin{pmatrix}y_n \\ a_n\end{pmatrix}\).
2. (b) Find the eigenvalues of \(M\).
3. (c) Find the long-term percentage increase in the population each year, and the long-term percentage of the seals that are young.

\(M = PDP^{-1}\). Show that \(M^3 = PD^3P^{-1}\), and explain why the same works for any \(n\).

The eigenvalues of a matrix \(M\) are 1 and 0.7. Explain why \(M^n\) settles to a fixed matrix as \(n\) grows. What would happen if they were 1 and 1.3?