If A = p q r r p q q r p and A A T = I , then p 3 + q 3 + r 3 =

It is given that, $A{A}^{\mathit{\top }}=I$

It is an orthogonal matrix and the determinant of orthogonal
matrix is $\pm 1 .$ .

$\left|\begin{array}{lll}p& q& r\\ r& p& q\\ q& r& p\end{array}\right|=\pm 1$

Solve the above determinant,

$p^3-pqr-pqr-pqr+q^3+r^3=\pm 1$

$\Rightarrow p^3+q^3+r^3-3pqr=\pm 1$

$\Rightarrow p^3+q^3+r^3=3pqr\pm 1$