Eigenvalues and eigenvectorsAQA A-Level Further Maths: Flashcards

What these 13 flashcards ask

Define an eigenvector and its eigenvalue.
What is the characteristic equation of M?
Characteristic equation of a 2\times2 matrix in terms of trace and determinant?
How do you check eigenvalues of a 2\times2 matrix?
How do you find the eigenvector for a given \lambda?
Why can an eigenvector only be found up to a multiple?
Why must an eigenvector not be the zero vector?
What does an eigenvector represent geometrically?
What does the eigenvalue represent geometrically?
What does a negative eigenvalue mean?
What does an eigenvalue of 1 mean?
What type of equation is the characteristic equation of a 3\times3 matrix?
Eigenvalues of \begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}?

Exam questions on Eigenvalues and eigenvectors

The matrix $M=\begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}$ .
Find an eigenvector of $M$ corresponding to the eigenvalue $1$ .2 marks
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
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

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).