Invariant points and linesAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

What is an invariant point?
How do you find invariant points of \mathbf{M}?
Which point is always invariant under a 2\times2 matrix?
What is a line of invariant points?
What is an invariant line?
Is every invariant line a line of invariant points?
How do you test whether y=mx is invariant under \begin{pmatrix}a&b\\ c&d\end{pmatrix}?
What quadratic gives invariant lines y=mx?
Why test x=0 separately?
How do you test y=mx+k with k\neq0?
Invariant lines of a reflection in a line through the origin?
Invariant lines of an enlargement with centre the origin?

Exam questions on Invariant points and lines

The transformation $T$ of the plane has matrix $\mathbf{M}=\begin{pmatrix}1&2\\0&3\end{pmatrix}$ .
Show that the line $y=2x$ is not invariant under $T$ .2 marks
The matrix $\mathbf{N}=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$ represents a reflection in the $y$ -axis.
Find the equation of the image of the line $y=2x+3$ under the reflection, and hence state whether this line is invariant.2 marks
The matrix $\mathbf{S}=\begin{pmatrix}2&-1\\1&0\end{pmatrix}$ represents a transformation $S$ of the plane.
Find the invariant points of $S$ .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).