Eigenvalues and eigenvectorsEdexcel A-Level Further Maths: Subtopic test

1. 1
   The matrix $\mathbf{A}=\begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}$.

   (a)

   Which is the characteristic equation of $\mathbf{A}$?

   [1 mark]

   • A$\lambda^2-5\lambda+4=0$
   • B$\lambda^2+5\lambda+4=0$
   • C$\lambda^2-5\lambda+8=0$
   • D$\lambda^2-4\lambda+5=0$

   (b)

   Which is an eigenvector of $\mathbf{A}$ corresponding to the eigenvalue $4$?

   [1 mark]

   • A$\begin{pmatrix} 1 \\ -1 \end{pmatrix}$
   • B$\begin{pmatrix} 1 \\ 2 \end{pmatrix}$
   • C$\begin{pmatrix} 1 \\ 1 \end{pmatrix}$
   • D$\begin{pmatrix} 2 \\ 1 \end{pmatrix}$

   (c)

   Find an eigenvector of $\mathbf{A}$ corresponding to the eigenvalue $1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{B}=\begin{pmatrix} 1 & -2 \\ 2 & 1 \end{pmatrix}$.

   (a)

   Find the eigenvalues of $\mathbf{B}$.

   [1 mark]

   • A$\pm2i$
   • B$1\pm2i$
   • C$1\pm4i$
   • D$1\pm\sqrt5$

   (b)

   Which is an eigenvector of $\mathbf{B}$ corresponding to the eigenvalue $1+2i$?

   [1 mark]

   • A$\begin{pmatrix} 1 \\ i \end{pmatrix}$
   • B$\begin{pmatrix} 1 \\ 1 \end{pmatrix}$
   • C$\begin{pmatrix} 2 \\ 1 \end{pmatrix}$
   • D$\begin{pmatrix} 1 \\ -i \end{pmatrix}$

   (c)

   Find an eigenvector of $\mathbf{B}$ corresponding to the eigenvalue $1-2i$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{C}=\begin{pmatrix} 3 & -1 \\ 1 & 1 \end{pmatrix}$.

   (a)

   Show that $\mathbf{C}$ has a repeated eigenvalue and find its value.

   [3 marks]

   (b)

   Find a normalised eigenvector of $\mathbf{C}$ corresponding to this eigenvalue.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $\mathbf{M}=\begin{pmatrix} a & 2 \\ 3 & 4 \end{pmatrix}$, where $a$ is a constant, has $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$ as an eigenvector.

   (a)

   Find the value of $a$ and the eigenvalue corresponding to the given eigenvector. Hence find the other eigenvalue of $\mathbf{M}$ and a corresponding eigenvector.

   [6 marks]

   (b)

   Using your answers to part (a), find $\mathbf{M}^4\begin{pmatrix} 4 \\ -1 \end{pmatrix}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions