A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin O and a set of x-y-axes. In each case, the drone moves to a new position represented by the following transformations:

(i) rotation anticlockwise $\frac{\pi}{6}$ is $\left(\begin{array}{cc}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right)$ (A1)

(ii) reflection in $y = \frac{x}{\sqrt{3}}$ is $\left(\begin{array}{cc}\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2}\end{array}\right)$ (A1)

(iii) rotation clockwise $\frac{\pi}{3}$ is $\left(\begin{array}{cc}\frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2}\end{array}\right)$ (A1)

an attempt to multiply three matrices

$\boldsymbol{P}=\left(\begin{array}{cc} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{array}\right)\left(\begin{array}{cc} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{array}\right)\left(\begin{array}{cc} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{array}\right)$ (M1) $\boldsymbol{P}=\left(\begin{array}{cc} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2} \end{array}\right) \quad \text{OR} \quad \left(\begin{array}{cc} 0.866 & -0.5 \\ -0.5 & -0.866 \end{array}\right)$ A1A1

$\boldsymbol{P}^{2}=\left(\begin{array}{cc}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2}\end{array}\right)\left(\begin{array}{cc}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2}\end{array}\right)=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$ (M1)A1 Note: Do not award $\boldsymbol{A1}$ if final answer not resolved into the identity matrix $I$.