Symmetric Matrices Using Pythagorean Triples 
Find symmetric matrices A =$\begin{pmatrix} a &b \\   c&d 
\end{pmatrix}$ such that $A^{2}=I_{2}$.

Alright, so I've posed this problem earlier but my question is in regard to this problem.
I was told that $\frac{1}{t}\begin{pmatrix}\mp r & \mp s \\   \mp s & \pm r  \end{pmatrix}$ is a valid matrix $A$ as $A^{2}=I_{2}$, given the condition that $r^{2}+s^{2}=t^{2}$, that is, (r,s,t) is a Pythagorean Triple.
Does anybody know why this works?

It works because $$A^2 = \frac{1}{t^2}\begin{pmatrix}r & s\\s & -r\end{pmatrix}\begin{pmatrix}r&s\\s&-r\end{pmatrix} = \frac{1}{t^2}\begin{pmatrix}r^2+s^2 & 0\\0 & r^2 + s^2\end{pmatrix}.$$
and you want the diagonals to be 1, i.e. $\frac{r^2 + s^2}{t^2} = 1$.  
