How to find $R_{ll}$ of the orthogonal matrix R?

Question

I get that $R_{kk}=\sqrt{1-s^2}$ but for $ R_{ll} $ I need to find $R_{lk} $ which is not given, so I am wondering how to find it? Or, is there a relation between $R_{kl} $ and $R_{lk} $ which I am missing?

Any help would be much appreciated!

Answer 1

We have two forms of orthogonal $2 \times 2$ matrices

$$\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}\text{ (rotation), }\qquad \begin{bmatrix} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \\ \end{bmatrix}\text{ (reflection)}$$

From this we can see what are possible relations between different entries of the orthogonal matrix. In the conditions of the task is that both diagonal entries are positive so they have to have the same sign... $(R_{kk}=R_{ll}$ and $R_{kl}=-R_{lk})$ ...