Let $A=\left[\begin{array}{lll}x & y & z \ y & z & x \ z & x & y\end{array} ight]$, where $x, y$

Question

Let $A=\left[\begin{array}{lll}x & y & z \ y & z & x \ z & x & y\end{array}ight]$, 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

Professor PC Thomas Classes · Answer

(7) : Given, $A^{2}=I$\ $\Rightarrow A A^{\prime}=I\left(\because A^{\prime}=Aight) \Rightarrow A$ is orthogonal.\ So, $x^{2}+y^{2}+z^{2}=1$ and $x y+y z+z x=0$\ Now, $(x+y+z)^{2}=\left(x^{2}+y^{2}+z^{2}ight)+2(x y+y z$ $+z x)=1+2(0)$\ $\Rightarrow \quad x+y+z=1 \quad[\because x+y+z>0]$\ Hence, $x^{3}+y^{3}+z^{3}$\ $=3 x y z+(x+y+z)\left(x^{2}+y^{2}+z^{2}-x y-y z-z xight)$ $=3 	imes 2+1(1-0)=7$\