Question

If a+b+c=0 them solve the following equation.
$\left|\begin{array}{ccc}a-x& c& b\\ c& b-x& a\\ b& a& c-x\end{array}\right|=0$

Answer 1

Given $\left|\begin{array}{ccc}a-x& c& b\\ c& b-x& a\\ b& a& c-x\end{array}\right|=0$

$R_1\to R_1+R_2+R_3$

$\left|\begin{array}{ccc}a+b+c-x& a+b+c-x& a+b+c-x\\ c& b-x& a\\ b& a& c-x\end{array}\right|=0$

$\left|\begin{array}{ccc}-x& -x& -x\\ c& b-x& a\\ b& a& c-x\end{array}\right|=0$

$(-x)\left|\begin{array}{ccc}1& 1& 1\\ c& b-x& a\\ b& a& c-x\end{array}\right|=0$

$x=0$ or $\left|\begin{array}{ccc}1& 1& 1\\ c& b-x& a\\ b& a& c-x\end{array}\right|=0$

$⟹1\left(\right(b-x\left)\right(c-x)-a^2)-1\left(c\right(c-x)-ab)+1(ac-b(b-x\left)\right)=0$

$⟹bc-(b+c)x+x^2-a^2-c^2+cx+ab+ac-b^2+bx=0$

$⟹x^2=a^2+b^2+c^2-ab-bc-ca$