Let Z 1 , Z 2 , Z 3 be three non zero complex numbers such that a = Z 1 , b = Z 2 , c = Z 3 . if the…

Let $Z_1,Z_2,Z_3$ be three non zero complex numbers such that $a=\left|Z_1\right|,b=\left|Z_2\right|,c=\left|Z_3\right|$.

Given,  $\left|\begin{array}{lll}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0$

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

$abc-a^3-b^3+abc+abc-c^3=0$

$3abc-a^3-b^3-c^3=0$

$a^3+b^3+c^3-3abc=0$

We know that:$a^3+b^3+c^3-3abc=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)$

$\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0$

Multiplying and dividing by $2$

$\frac{1}{2}\left(a+b+c\right)\left(2a^2+2b^2+2c^2-2ab-2bc-2ca\right)=0$

$\frac{1}{2}\left(a+b+c\right)\left[{\left(a-b\right)}^2+{\left(b-c\right)}^2+{\left(c-a\right)}^2\right]=0$

$a+b+c=0$

Hence, $\left|z_1\right|+\left|z_2\right|+\left|z_3\right|=0$