What should be the value of $p,q,r,s,$ and $t$ so that the following matrix is orthogonal?
$$\left[ {\begin{array}{cc} p & q & r \\ \frac{1}{\sqrt{3}} & q & s \\ \frac{1}{\sqrt{3}} & 0 & t \\ \end{array} } \right]$$
I tried to solve this problem and here's what I got.
Let the given matrix be matrix $A$. Hence,
$$AA^{t}=I$$
So,
$$ \left[ {\begin{array}{cc} p & q & r \\ \frac{1}{\sqrt{3}} & q & s \\ \frac{1}{\sqrt{3}} & 0 & t \\ \end{array} } \right] \left[ {\begin{array}{cc} p & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ q & q & 0 \\ r & s & t \\ \end{array} } \right] = \left[ {\begin{array}{cc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]$$
Here are the equations I got.
$$p^2 + q^2 +r^2 = 1$$ $$ q^2 + s^2 = \frac{2}{3}$$ $$ t^2=\frac{2}{3}$$
$$\frac{1}{\sqrt{3}}p+q^2+rs=0$$ $$\frac{1}{\sqrt{3}}p+tr=0$$ $$st=-\frac{1}{3}$$
After this, I am stuck in this problem. I can only find the value of $t$ and $s$.