How do you find the invariant points of \mathbf{M}?

Solve \mathbf{M}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}x\\ y\end{pmatrix} for x and y.

What is a line of invariant points?

A line on which every point is mapped to itself, e.g. the mirror line of a reflection.

What is an invariant line?

A line whose points all map to points on the same line.

Is every invariant line a line of invariant points?

No. Points may move along the line; only the whole-line-fixed case is a line of invariant points.

How do you test whether y=mx+c is invariant?

Map the general point (x,mx+c) and check the image satisfies y'=mx'+c for all x.

Why check x=0 separately?

It has no gradient, so the equation for m cannot find it.

Invariant points and linesEdexcel A-Level Further Maths: Flashcards

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What is an invariant point?

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What is an invariant point?: A point mapped to itself: $\mathbf{M}\mathbf{x}=\mathbf{x}$ .
How do you find the invariant points of $\mathbf{M}$ ?: Solve $\mathbf{M}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}x\\ y\end{pmatrix}$ for $x$ and $y$ .
What is a line of invariant points?: A line on which every point is mapped to itself, e.g. the mirror line of a reflection.
Which point is always invariant for a matrix transformation?: The origin, $(0,0)$ .
What is an invariant line?: A line whose points all map to points on the same line.
Is every invariant line a line of invariant points?: No. Points may move along the line; only the whole-line-fixed case is a line of invariant points.
How do you test whether $y=mx+c$ is invariant?: Map the general point $(x,mx+c)$ and check the image satisfies $y'=mx'+c$ for all $x$ .
For $\mathbf{M}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ , what equation gives invariant lines $y=mx$ ?: $c+dm=m(a+bm)$
Why check $x=0$ separately?: It has no gradient, so the equation for $m$ cannot find it.
Invariant points of reflection in the line $y=x$ ?: Every point on $y=x$ .
Invariant points of an enlargement, centre the origin, scale factor $2$ ?: Only the origin.
Invariant lines through the origin for an enlargement?: Every line through the origin.

Exam questions on Invariant points and lines

The matrix $\mathbf{A}=\begin{pmatrix}3&-2\\ 1&0\end{pmatrix}$ represents a linear transformation of the plane.
Find the coordinates of all the invariant points of the transformation.2 marks
The matrix $\mathbf{M}=\begin{pmatrix}2&0\\ 0&3\end{pmatrix}$ represents a linear transformation of the plane.
Find the equations of all the invariant lines through the origin.2 marks
The matrix $\mathbf{B}=\begin{pmatrix}4&2\\ 1&3\end{pmatrix}$ represents a linear transformation of the plane.
Show that the invariant lines through the origin of the form $y=mx$ satisfy $2m^2+m-1=0$ .3 marks

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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).