The value of `|(a-b,b+c,a),(b-a,c+a,b),(c-a,a+b,c)|` is

Options

A. `a^(3)+b^(3)+c^(3)`
B. `3bc`
C. `a^(3)+b^(3)+c^(3)-3abc`
D. None of these

Answer

D

Explanation

We have
`|(a-b,b+c,a),(b-a,c+a,b),(c-a,a+b,c)|=|(a+c,b+c+a,a),(b+c,c+a+b,b),(c+b,a+b+c,c)| [ :' C_(1)toC_(1)+C_(2)` and `C_(2)toC_(2)+C_(3)]`
`=(a+b+c)|(a+c,1,a),(b+c,1,b),(c+b,1,c)|` [taking `(a+b+c)` common from `C_(2)`]
`=(a+b+c)|(a-b,0,a-c),(0,0,b-c),(c+b,1,c)| [ :' R_(2)toR_(2)-R_(3)` and `R_(1)toR_(1)-R_(3)]`
`=(a+b+c)[-(b-c).(a-b)]` [expanding along `R_(2)`]
`=(a+b+c)(c-b)(a-b)`