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Eigenvalues and eigenvectorsEdexcel A-Level Further Maths: Mind map

Definition
Characteristic equation
Finding vectors

Eigenvalues and eigenvectors

2 x 2 matrices

Av=λv\mathbf{A}\mathbf{v}=\lambda\mathbf{v}characteristic equationnormalised
Repeated roots
Complex roots
Applications

Exam questions on Eigenvalues and eigenvectors

  1. The matrix A=(3122)\mathbf{A}=\begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}.
    Find an eigenvector of A\mathbf{A} corresponding to the eigenvalue 11.2 marks
  2. The matrix B=(1−221)\mathbf{B}=\begin{pmatrix} 1 & -2 \\ 2 & 1 \end{pmatrix}.
    Find an eigenvector of B\mathbf{B} corresponding to the eigenvalue 1−2i1-2i.2 marks
  3. The matrix C=(3−111)\mathbf{C}=\begin{pmatrix} 3 & -1 \\ 1 & 1 \end{pmatrix}.
    Show that C\mathbf{C} has a repeated eigenvalue and find its value.3 marks
See the full worksheet

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).