Eigenvalues and eigenvectorsAQA A-Level Further Maths: Subtopic test

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

   (a)

   Which equation is the characteristic equation of $M$?

   [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 of these is an eigenvector of $M$?

   [1 mark]

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

   (c)

   Find an eigenvector of $M$ corresponding to the eigenvalue $1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $A=\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$ represents a linear transformation of the plane.

   (a)

   Find the eigenvalues of $A$.

   [1 mark]

   • A$1$ and $2$
   • B$-3$ and $1$
   • C$3$ and $-1$
   • D$3$ and $1$

   (b)

   Which line through the origin is mapped to itself with the position vector of every point on it reversed in direction?

   [1 mark]

   • A$y=-x$
   • B$y=x$
   • C$y=2x$
   • D$y=-2x$

   (c)

   Describe the geometrical effect of $A$ on a point on the line $y=x$.

   [2 marks]

   Total for question 2: 4 marks

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

   (a)

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

   [3 marks]

   (b)

   Find the other eigenvalue of $A$ and a corresponding eigenvector.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $Q=\begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}$.

   (a)

   Show that the eigenvalues of $Q$ satisfy $(2-\lambda)\left((2-\lambda)^2-2\right)=0$, and hence find the exact eigenvalues of $Q$.

   [6 marks]

   (b)

   Find an eigenvector of $Q$ for each of its three eigenvalues.

   [6 marks]

   Total for question 4: 12 marks

End of questions