Eigenvalues and eigenvectors Problem set A

Find the eigenvalues and eigenvectors of each matrix.

1. (a) \(\begin{pmatrix}3 & 2 \\ 1 & 4\end{pmatrix}\)
2. (b) \(\begin{pmatrix}1 & 2 \\ 4 & 3\end{pmatrix}\)
3. (c) \(\begin{pmatrix}6 & -2 \\ 2 & 1\end{pmatrix}\)
4. (d) \(\begin{pmatrix}0.8 & 0.3 \\ 0.2 & 0.7\end{pmatrix}\)

Find the characteristic polynomial of \(\begin{pmatrix}5 & 4 \\ 2 & -2\end{pmatrix}\), and hence its eigenvalues and eigenvectors.

\(\begin{pmatrix}2 \\ 1\end{pmatrix}\) is an eigenvector of \(\begin{pmatrix}4 & a \\ 1 & 3\end{pmatrix}\). Find \(a\) and both eigenvalues.

Show that \(\begin{pmatrix}1 & -3 \\ 2 & 1\end{pmatrix}\) has no real eigenvalues.

Find the eigenvalue and eigenvector of \(\begin{pmatrix}3 & 1 \\ 0 & 3\end{pmatrix}\). What is unusual?

\(M = \begin{pmatrix}k & 1 \\ 4 & 3\end{pmatrix}\) has an eigenvalue \(5\).

1. (a) Find the value of \(k\).
2. (b) Find the other eigenvalue.
3. (c) Find an eigenvector for the eigenvalue \(1\).

A matrix has eigenvectors \(\begin{pmatrix}1 \\ 2\end{pmatrix}\) and \(\begin{pmatrix}1 \\ -1\end{pmatrix}\) with eigenvalues 4 and 1. Find the matrix.

Show that the eigenvalues of \(\begin{pmatrix}p & q \\ q & p\end{pmatrix}\) are \(p + q\) and \(p - q\), and find the eigenvectors.