Determine geometric results by matrix multiplications, e.g. showing that the combined effect of two reflections in lines through the origin is a rotation.

A reflection in the line \(y = x\tan(\theta)\) through the origin can be represented by the matrix \(R_{\theta} = \begin{bmatrix} \cos(2\theta) & \sin(2\theta) \\ \sin(2\theta) & -\cos(2\theta) \end{bmatrix}\). Consider the reflection in the line \(y = x\) followed by the reflection in the line \(y = \sqrt{3}x\). Determine the single transformation equivalent to this combined transformation by finding the product of the two reflection matrices. State the angle of rotation and direction.

Let \(R_1\) represent reflection in the line \(y = x\) through the origin, and let \(R_2\) represent reflection in the \(x\)-axis. The matrix representing \(R_1\) is \(M_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) and the matrix representing \(R_2\) is \(M_2 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\). **a)** Determine the matrix \(M\) representing the combined transformation \(R_2\) followed by \(R_1\). **(2 marks)** **b)** Determine the single geometric transformation that \(M\) represents. **(2 marks)**

Consider the reflection matrix \(R_1\) representing reflection in the line \(y = x\sqrt{3}\) and the reflection matrix \(R_2\) representing reflection in the line \(y = 0\) (the \(x\)-axis), as shown in the table below. Determine the angle of rotation, \(\theta\), where \(0^\circ \leq \theta < 360^\circ\), that results from the combined transformation \(R_1R_2\).

Matrix \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) represents a reflection in the line \(y = x\), and matrix \(B = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) represents a reflection in the \(x\)-axis. Determine the single transformation represented by the combined transformation \(BA\).