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IB Mathematics: Applications and Interpretation HL · AHL 5: Matrices · AHL 5.6

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Eigenvalues and eigenvectors Problem set B

Work in your exercise book. Check each eigenvector by multiplying it by the matrix.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. Find the eigenvalues and eigenvectors of each matrix.

    1. (a) \(\begin{pmatrix}6 & 1 \\ 2 & 5\end{pmatrix}\)
    2. (b) \(\begin{pmatrix}1 & 3 \\ 2 & 2\end{pmatrix}\)
    3. (c) \(\begin{pmatrix}7 & -3 \\ 2 & 2\end{pmatrix}\)
    4. (d) \(\begin{pmatrix}0.7 & 0.4 \\ 0.3 & 0.6\end{pmatrix}\)

  2. Find the characteristic polynomial of \(\begin{pmatrix}4 & 3 \\ 2 & -1\end{pmatrix}\), and hence its eigenvalues and eigenvectors.

  3. \(\begin{pmatrix}1 \\ 1\end{pmatrix}\) is an eigenvector of \(\begin{pmatrix}3 & a \\ 2 & 6\end{pmatrix}\). Find \(a\) and both eigenvalues.

Thinking

  1. Show that \(\begin{pmatrix}1 & -4 \\ 2 & 1\end{pmatrix}\) has no real eigenvalues.

  2. Find the eigenvalue and eigenvector of \(\begin{pmatrix}5 & 2 \\ 0 & 5\end{pmatrix}\). What is unusual?

  3. \(M = \begin{pmatrix}k & 2 \\ 3 & 4\end{pmatrix}\) has an eigenvalue \(7\).

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

Stretch

  1. A matrix has eigenvectors \(\begin{pmatrix}1 \\ 1\end{pmatrix}\) and \(\begin{pmatrix}1 \\ 2\end{pmatrix}\) with eigenvalues 3 and −1. Find the matrix.

  2. Show that the eigenvalues of \(\begin{pmatrix}p & 4q \\ q & p\end{pmatrix}\) are \(p + 2q\) and \(p - 2q\), and find the eigenvectors.

jmaths.xyz · AI HL · AHL 5.6 Eigenvalues and eigenvectors