Diagonalisation of matricesAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

What is diagonalisation of M?
What are the columns of U?
What is on the diagonal of D?
What is M^n in terms of U and D?
Why does M^2=UD^2U^{-1}?
How do you raise a diagonal matrix to the power n?
What are the eigenvalues of M^n?
Is the matrix U unique?
Inverse of \begin{pmatrix} a & b \\ c & d \end{pmatrix}?
How can you check M=UDU^{-1} without finding U^{-1}?
Can a matrix with no real eigenvalues be diagonalised using real matrices?
Quick check of a formula for M^n?

Exam questions on Diagonalisation of matrices

The matrix $M=\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$ has eigenvalues $2$ and $5$ , with corresponding eigenvectors $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ .
Taking $U=\begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}$ , find $U^{-1}$ .2 marks
The matrix $A=\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ can be written as $A=UDU^{-1}$ , where $U=\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ and $D$ is a diagonal matrix.
Explain why $A^4$ has the same eigenvectors as $A$ , and state the eigenvalues of $A^4$ .2 marks
The matrix $B=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 3 & 1 \\ 0 & 1 & 3 \end{pmatrix}$ has eigenvalues $1$ , $2$ and $4$ , with corresponding eigenvectors $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$ , $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$ .
Write down a matrix $U$ and a diagonal matrix $D$ such that $B=UDU^{-1}$ .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).