Question

If A = $\begin{bmatrix}p&q&r\\ r&p&q\\ q&r&p\end{bmatrix} $ and $AA^T = I$ 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 $\pm 1$.
$\therefore |A|=\begin{vmatrix}p & q & r \\ r & p & q \\ q & r & p\end{vmatrix}=\pm 1$
$\Rightarrow p\left(p^{2}-q r\right)-q\left(p r-q^{2}\right)+r\left(r^{2}-p q\right)=\pm 1$
$\Rightarrow p^{3}-p q r-p q r+q^{3}+r^{3}-p q r=\pm 1$
$\Rightarrow p^{3}+q^{3}+r^{3}-3 p q r=\pm 1$
$\Rightarrow p^{3}+q^{3}+r^{3}=3 p q r \pm 1$