Eigenvalues and eigenvectors – university Year 1 maths test to download as PDF

Find the eigenvalues and corresponding eigenvectors of \(\begin{pmatrix}1&2\\2&1\end{pmatrix}\).

Let \(A=\) \(\begin{pmatrix}3&1\\0&2\end{pmatrix}\). Find \(P\) and \(D\) with \(A=PDP^{-1}\) and hence \(A^n\).

1. Find the eigenvalues of \(\begin{pmatrix}0&-4\\1&0\end{pmatrix}\). (2 marks)
2. Show that \(\begin{pmatrix}1&1\\0&1\end{pmatrix}\) has only one independent eigenvector. (2 marks)

Each year 20% of cinema-goers switch to streaming and 10% of streaming users go back to the cinema. The transition matrix on (cinema, streaming) is \(\begin{pmatrix}0.8&0.1\\0.2&0.9\end{pmatrix}\).

1. Find the eigenvalues. (2 marks)
2. Find the long-term proportion using cinema. (2 marks)

For \(A=\) \(\begin{pmatrix}1&2\\3&0\end{pmatrix}\), show that \(A^2=A+6I\) and use this to find \(A^{-1}\).