Problem
Q4. Styles \[ \mathbf{M}=\left(\begin{array}{cc} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{array}\right) \] (a) Show that $\mathbf{M}$ is non-singular. (2) The hexagon $R$ is transformed to the hexagon $S$ by the transformation represented by the matrix $M$. Given that the area of hexagon $R$ is 5 square units, (b) find the area of hexagon $S$. (1) The matrix $\mathbf{M}$ represents an enlargement, with centre $(0,0)$ and scale factor $k$, where $k>0$, followed by a rotation anti-clockwise through an angle $\theta$ about $(0,0)$. (c) Find the value of $k$. (2) (d) Find the value of $\theta$. (2) (Total for question $=7$ marks)
Solution
Step 1 ：\(\text{det}(\mathbf{M}) = \left|\begin{array}{cc} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{array}\right| = 1(1) - (-\sqrt{3})(\sqrt{3}) = 1 + 3 = 4 \neq 0\)
Step 2 ：\(\text{Area of hexagon S} = \text{Area of hexagon R} \times |\text{det}(\mathbf{M})| = 5 \times 4 = \boxed{20}\)
Step 3 ：\(k = \sqrt{\left(\sqrt{3}\right)^2 + 1^2} = \sqrt{3 + 1} = \boxed{2}\)
Step 4 ：\(\cos\theta = \frac{1}{2}\), \(\theta = \cos^{-1}\left(\frac{1}{2}\right) = \boxed{60^\circ}\)
