determine the 2 × 2 matrix P which represents a reflection in the line $y = \left( {{\text{tan}}\,\theta } \right)x$.

determine the 2 × 2 matrix Q which represents an anticlockwise rotation of θ about the origin.

Describe the transformation represented by the matrix PQ.

A matrix M is said to be orthogonal if M ^TM = I where I is the identity. Show that Q is orthogonal.

      (M1)

using the transformation of the unit square:

$\left( \begin{gathered} 1 \hfill \\ 0 \hfill \\ \end{gathered} \right) \to \left( {\begin{array}{*{20}{c}} {{\text{cos}}\,2\theta } \\ {{\text{sin}}\,2\theta } \end{array}} \right)$ and $\left( \begin{gathered} 0 \hfill \\ 1 \hfill \\ \end{gathered} \right) \to \left( {\begin{array}{*{20}{c}} {{\text{sin}}\,2\theta } \\ { - {\text{cos}}\,2\theta } \end{array}} \right)$       (M1)

hence the matrix P is $\left( {\begin{array}{*{20}{c}} {{\text{cos}}\,2\theta } \\ {{\text{sin}}\,2\theta } \end{array}\,\,\,\,\begin{array}{*{20}{c}} {{\text{sin}}\,2\theta } \\ { - {\text{cos}}\,2\theta } \end{array}} \right)$       A1

[3 marks]

using the transformation of the unit square:

$\left( \begin{gathered} 1 \hfill \\ 0 \hfill \\ \end{gathered} \right) \to \left( {\begin{array}{*{20}{c}} {{\text{cos}}\,\theta } \\ {{\text{sin}}\,\theta } \end{array}} \right)$ and $\left( \begin{gathered} 0 \hfill \\ 1 \hfill \\ \end{gathered} \right) \to \left( {\begin{array}{*{20}{c}} { - {\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right)$      (M1)

hence the matrix Q is $\left( {\begin{array}{*{20}{c}} {{\text{cos}}\,\theta } \\ {{\text{sin}}\,\theta } \end{array}\,\,\,\,\,\,\begin{array}{*{20}{c}} { - {\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right)$      A1

[2 marks]

PQ = $\left( {\begin{array}{*{20}{c}} {{\text{cos}}\,\theta \,{\text{cos}}\,2\theta + {\text{sin}}\,\theta \,{\text{sin}}\,2\theta } \\ { - {\text{cos}}\,2\theta \,{\text{sin}}\,\theta + \,{\text{sin}}\,2\theta \,{\text{cos}}\,\theta } \end{array}\,\,\,\,\,\,\begin{array}{*{20}{c}} {{\text{cos}}\,\theta \,{\text{sin}}\,2\theta \, - {\text{sin}}\,\theta \,{\text{cos}}\,2\theta } \\ { - {\text{sin}}\,\theta \,{\text{sin}}\,2\theta - {\text{cos}}\,\theta \,{\text{cos}}\,2\theta \,} \end{array}} \right)$      M1A1

$= \left( {\begin{array}{*{20}{c}} {{\text{cos}}\,\left( {2\theta - \theta } \right)} \\ {{\text{sin}}\,\left( {2\theta - \theta } \right)} \end{array}\,\,\,\,\,\,\begin{array}{*{20}{c}} {{\text{sin}}\,\left( {2\theta - \theta } \right)} \\ { - {\text{cos}}\,\left( {2\theta - \theta } \right)} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{\text{cos}}\,\theta } \\ {{\text{sin}}\,\theta } \end{array}\,\,\,\,\,\,\begin{array}{*{20}{c}} {{\text{sin}}\,\theta } \\ { - {\text{cos}}\,\theta } \end{array}} \right)$     M1A1

this is a reflection in the line $y = \left( {{\text{tan}}\,\frac{1}{2}\theta } \right)x$      A1

[5 marks]

Q ^TQ = $\left( {\begin{array}{*{20}{c}} {{\text{cos}}\,\theta } \\ { - {\text{sin}}\,\theta } \end{array}\,\,\,\,\,\,\begin{array}{*{20}{c}} {{\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{\text{cos}}\,\theta } \\ {{\text{sin}}\,\theta } \end{array}\,\,\,\,\,\,\begin{array}{*{20}{c}} { - {\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right)$

$= \left( {\begin{array}{*{20}{c}} {{\text{co}}{{\text{s}}^2}\,\theta + {\text{si}}{{\text{n}}^2}\,\theta } \\ { - {\text{sin}}\,\theta \,{\text{cos}}\,\theta + {\text{cos}}\,\theta \,{\text{sin}}\,\theta } \end{array}\,\,\,\,\,\,\begin{array}{*{20}{c}} { - {\text{cos}}\,\theta \,{\text{sin}}\,\theta + {\text{cos}}\,\theta \,{\text{sin}}\,\theta } \\ {{\text{si}}{{\text{n}}^2}\,\theta + {\text{co}}{{\text{s}}^2}\,\theta } \end{array}} \right)$      M1A1

$= \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)$      AG

[2 marks]