1. Problem: Write the transformation matrix of rotation about origin through an angle 45° in clockwise direction. 2. Formula: The rotation matrix for an angle $\theta$ in clockwise direction is given by: $$\begin{bmatrix}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$$ 3. Calculation: For $\theta = 45^\circ$, convert to radians $\theta = \frac{\pi}{4}$. 4. Substitute values: $$\cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}$$ 5. So the matrix is: $$\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}$$ This is the required rotation matrix. q_count is 3 as there are 3 distinct questions in the user message.