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.

Explanation

`L.H.S. = |[a,b,c],[b,c,a],[c,a,b]|`

Applying `C_1->C_1+C_2+C_3`

` = |[a+b+c,b,c],[a+b+c,c,a],[a+b+c,a,b]|`

` = (a+b+c) |[1,b,c],[1,c,a],[1,a,b]|`

Applying `R_2->R_2-R_1` and `R_3->R_3-R_1`

` = (a+b+c)|[1,b,c],[0,c-b,a-c],[0,a-b,b-c]|`

` = (a+b+c) [1((c-b)(b-c)-(a-b)(a-c))-0+0]`

` = -(a+b+c)[(b-c)^2+a^2-ab-ac+bc]`

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