Find all possible values of $p$, $q$ and $r$ such that this matrix is orthogonal 
Find all possible values of $p$, $q$ and $r$ such that the following matrix is orthogonal.
$$B= \begin{pmatrix}
\frac1{3}&\frac2{3}&\frac2{3} \\
\frac2{3}&\frac1{3}&-\frac2{3} \\
p&q&r
\end{pmatrix}$$


If $B$ is your matrix, then$$B^T.B=\begin{pmatrix}p^2+\frac{5}{9} & p q+\frac{4}{9} & p r-\frac{2}{9}   \\ p q+\frac{4}{9} & q^2+\frac{5}{9} & q r+\frac{2}{9}  \\ p r-\frac{2}{9} & q r+\frac{2}{9} & r^2+\frac{8}{9}\end{pmatrix}$$and you want this to be the identity matrix. So, since you only want to have $1$'s in the main diagonal, $p=q=\pm\frac23$ and $r=\pm\frac13$. Can you take it from here?
