Question

If A = $\left[\begin{array}{ccc}p& q& r\\ r& p& q\\ q& r& p\end{array}\right]$ and $A{A}^T=I$ then, $p^3+q^3+r^3$ =

• A. $\pm 1$
• B. $pqr$
• C. $3pqr$
• D. $3pqr\pm 1$

Answer 1

Answer: (D)

Given, $A{A}^T=I$
It represents orthogonal matrix.
Determinant of orthogonal matrix is $\pm 1$.
$\therefore |A|=\left|\begin{array}{ccc}p& q& r\\ r& p& q\\ q& r& p\end{array}\right|=\pm 1$
$\Rightarrow p\left(p^2-qr\right)-q\left(pr-q^2\right)+r\left(r^2-pq\right)=\pm 1$
$\Rightarrow p^3-pqr-pqr+q^3+r^3-pqr=\pm 1$
$\Rightarrow p^3+q^3+r^3-3pqr=\pm 1$
$\Rightarrow p^3+q^3+r^3=3pqr\pm 1$