If `a , b , c` are real numbers, prove that `|a b c b c a c a b|=-(a+b+c)(c+b w+c w^2)(a+b w^2+c w),` where `w` is a complex cube root of unity.

Answer

Proved

Explanation

$$ L. H. S. =\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$$
$$ \text { Applying } C_{1} \rightarrow C_{1}+C_{2}+C_{3}$$
$$
\begin{aligned}
&=\left|\begin{array}{lll}
a+b+c & b & c \\
a+b+c & c & a \\
a+b+c & a & b
...