Question

Let $A=\begin{bmatrix}x&y&z\\ y&z&x\\ z&x&y\end{bmatrix}$ where $x, y$ and $z$ are real numbers such that $x+y+z>0$ and $x y z=2$ If $A^{2}=I_{3}$, then the value of $x^{3}+y^{3}+z^{3}$ is

Answer 1

$A ^{2}= I$
$\Rightarrow AA'= I$
$\left(\right.$ as $\left. A ^{\prime}= A \right)$
$\Rightarrow A$ is orthogonal
So, $x^{2}+y^{2}+z^{2}=1$ and $x y+y z+z x=0$
$\Rightarrow (x+y+z)^{2}=1+2 \times 0 $
$\Rightarrow x+y+z=1$
Thus,
$x ^{3}+ y ^{3}+ z ^{3} =3 \times 2+1 \times(1-0)$
$=7$