Eigenvalues and eigenvectorsEdexcel International A Level Further Maths: Subtopic test

1. 1
   $\mathbf{A}=\begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$.

   (a)

   What are the eigenvalues of $\mathbf{A}$?

   [1 mark]

   • A$1$ and $12$
   • B$2$ and $5$
   • C$-2$ and $-5$
   • D$3$ and $4$

   (b)

   Which vector is an eigenvector of $\mathbf{A}$ corresponding to the eigenvalue $5$?

   [1 mark]

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

   (c)

   Find a normalised eigenvector of $\mathbf{A}$ corresponding to the eigenvalue $2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $\mathbf{B}=\begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 4 \\ 0 & 4 & 9 \end{pmatrix}$.

   (a)

   Which of the following is an eigenvalue of $\mathbf{B}$?

   [1 mark]

   • A$3$
   • B$4$
   • C$9$
   • D$11$

   (b)

   What is the product of the three eigenvalues of $\mathbf{B}$?

   [1 mark]

   • A$22$
   • B$14$
   • C$54$
   • D$11$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{M}=\begin{pmatrix} 5 & k \\ 1 & 2 \end{pmatrix}$, where $k$ is a constant, has $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ as an eigenvector.

   (a)

   Find the eigenvalue corresponding to this eigenvector, and the value of $k$.

   [3 marks]

   (b)

   Find the other eigenvalue of $\mathbf{M}$ and a corresponding eigenvector.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $\mathbf{C}=\begin{pmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 0 & 0 & 5 \end{pmatrix}$.

   (a)

   Find the eigenvalues of $\mathbf{C}$, and a normalised eigenvector corresponding to the largest eigenvalue.

   [6 marks]

   (b)

   Find an eigenvector of $\mathbf{C}$ corresponding to each of the other two eigenvalues, and verify one of them by multiplying it by $\mathbf{C}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions