Eigenvalues and eigenvectorsAQA A-Level Further Maths: Mind map

Exam questions on Eigenvalues and eigenvectors

1. The matrix $M=\begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}$.
   Find an eigenvector of $M$ corresponding to the eigenvalue $1$.2 marks
2. The matrix $A=\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$ represents a linear transformation of the plane.
   Describe the geometrical effect of $A$ on a point on the line $y=x$.2 marks
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.
   Find the eigenvalue corresponding to this eigenvector, and find the value of $k$.3 marks