Question

Let $A=\left[\begin{array}{ccc}x& y& z\\ y& z& x\\ z& x& y\end{array}\right]$, where $x,y$ and $z$ are real numbers such that $x+y+z>0$ a$xyz=2$.
If $A^2=I_3$, then the value of $x^3+y^3+z^3$ is

Answer 1

$A=\left[\begin{array}{ccc}x& y& z\\ y& z& x\\ z& x& y\end{array}\right]$ $\therefore |A|=(x^3+y^3+z^3-3xyz)$
$A^2=I_3$
$|A^2|=1$
$\therefore (x^3+y^3+z^3-3xyz{)}^2=1$
$\Rightarrow x^3+y^3+z^3-3xyz=1$ as $(x+y+z>0)$
$\Rightarrow x^3+y^3+z^3=6+1=7$