Diagonalisation and powers Problem set A

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

1. (a) \(\begin{pmatrix}3 & 2 \\ 1 & 4\end{pmatrix}\)
2. (b) \(\begin{pmatrix}1 & 2 \\ 4 & 3\end{pmatrix}\)

For \(M = \begin{pmatrix}3 & 2 \\ 1 & 4\end{pmatrix}\), find \(M^n\) in terms of \(n\), and hence find \(M^4\).

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

A matrix \(M\) has eigenvalue \(4\) with eigenvector \(\begin{pmatrix}1 \\ 2\end{pmatrix}\) and eigenvalue \(1\) with eigenvector \(\begin{pmatrix}1 \\ 1\end{pmatrix}\).

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

Each year 20% of the people living in a town move to the countryside around it, and 30% of the people in the countryside move into the town. This is modelled by \(T = \begin{pmatrix}0.8 & 0.3 \\ 0.2 & 0.7\end{pmatrix}\). At the start there are 5000 people in each. (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 beetle produces 2.2 young, 20% of the young survive to become adults, and 70% of the adults survive. Let \(y_n\) and \(a_n\) be the numbers of young and adult beetles 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 beetles that are young.

Explain why \(M^n = PD^nP^{-1}\) when \(M = PDP^{-1}\).

The eigenvalues of a matrix \(M\) are 1 and 0.4. Explain why \(M^n\) settles to a fixed matrix as \(n\) grows. What would happen if they were 1 and 1.4?