Edexcel IAL Mathematics FP1.6.1 column vector transformations

Practise applying 2 x 2 transformation matrices to column vectors, including solving for unknown coordinates from mapped points.

Exam points

multiply a matrix by a column vector and compare components with the mapped point
solve for an unknown parameter when a point is transformed to a stated image

[Maximum number: 5]

The transformation W is represented by the matrix

\left(\begin{array}{cc} -2 & 2 \sqrt{3} \\ 2 \sqrt{3} & 2 \end{array}\right)

Show that there is a real number $\lambda$ for which W maps the point $(\lambda, 1)$ onto the point ( $4 \lambda, 4$ ), giving the exact value of $\lambda$

$\begin{pmatrix}-2&2\sqrt3\\2\sqrt3&2\end{pmatrix}\binom{\lambda}{1}=\binom{4\lambda}{4}$ $\binom{-2\lambda+2\sqrt3}{2\sqrt3\lambda+2}=\binom{4\lambda}{4}$
M1
$-2\lambda+2\sqrt3=4\lambda\quad\text{or}\quad2\sqrt3\lambda+2=4$
A1
$6\lambda=2\sqrt3\Rightarrow\lambda=\frac{\sqrt3}{3}$
or $2\sqrt3\lambda=2\Rightarrow\lambda=\frac1{\sqrt3}$
dM1 A1
Both equations solved leading to $\lambda=\frac{\sqrt3}{3}$ or $\frac1{\sqrt3}$
A1

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FP1.6.1 - Linear transformations of column question 1