Question 6.4.25: Find all 2 by 2 matrices that are orthogonal and also symmet...

Orthogonal and symmetric requires \left| \lambda \right| = 1 and \lambda real, so \lambda = \pm 1.

Then A = \pm I or

A = Q\wedge {Q}^{T} =\begin{bmatrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{bmatrix}=\begin{bmatrix} \cos { 2\theta } & \sin { 2\theta } \\ \sin { 2\theta } & -\cos { 2\theta } \end{bmatrix}.