Let's answer **Question 1(a)** in 2 steps, with explanations and a final answer: --- ### 1(a) Write the matrix representation of a C₂ rotation about the z-axis. --- #### **Step 1: Understand What a C₂ Rotation Is** A **C₂ rotation about the z-axis** means a rotation of 180° (or π radians) around the z-axis. In matrix form, this rotation is represented in 3D Cartesian coordinates (x, y, z). - After a 180° rotation about the z-axis: - The x and y coordinates will invert (change sign). - The z coordinate remains unchanged. #### **Explanation:** - If you rotate a point (x, y, z) by 180° about the z-axis, the new coordinates are (-x, -y, z). --- #### **Step 2: Write the Matrix Representation** The transformation matrix for a 180° rotation about the z-axis is: \[ \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \] #### **Explanation:** - The matrix multiplies the x and y components by -1 (flips them), and leaves the z component unchanged. --- #### **Final Answer:** \[ \boxed{ \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} } \] This is the matrix representation of a C₂ rotation about the z-axis. --- Would you like me to continue with 1(b) in the same format?