CAIE A-Level Further Math AS 1.4.5 Invariant Points and Lines Questions

Practise finding invariant points and invariant lines under matrix transformations.

Exam points

identify invariant lines of matrix transformations
solve for invariant points and line equations
apply invariant line methods to transformations

[Maximum number: 5]

Let k be a constant. The matrices A, B and C are given by

\mathbf{A}=\left(\begin{array}{lll} 1 & k & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rr} 0 & -2 \\ -1 & 3 \\ 0 & 0 \end{array}\right) \quad \text { and } \quad \mathbf{C}=\left(\begin{array}{rrr} -2 & -1 & 1 \\ 1 & 1 & 3 \end{array}\right)

It is given that A is singular.

Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane represented by CAB.

$\left(\begin{array}{cc}3 -7 -9 3\end{array}\right)\binom{x}{y}=\binom{3 x-7 y}{-9 x+3 y}$
Transforms $\binom{x}{y}$ to $\binom{X}{Y}$ .

-9 x+3 m x=m(3 x-7 m x)
Uses y=m x and Y=m X.

$-9+3 m=3 m-7 m^{2} \Rightarrow 7 m^{2}=9$ $y=\frac{3}{\sqrt{7}} x$ and $y=-\frac{3}{\sqrt{7}} x$
5

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CAIE A-Level Further Math AS 1.4.5 Invariant Points and Lines Questions question 1