The determinat `Delta=|(b^2-ab,b-c,bc-ac),(ab-a^2,a-b,b^2-ab),(bc-ac,c-a,ab-a^2)|` equals

Options

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

Answer

D

Explanation

Weh have
`|(b^(2)-ab,b-c,bc-ac),(ab-a^(2),a-b,b^(2)-ab),(bc-ac,c-a,ab-a^(2))|=|(b(b-a),b-c,c(b-a)),(a(b-a),a-b,b(b-a)),(c(b-a),c-a,a(b-a))|`
`=(b-a)^(2)|(b,b-c,c),(a,a-b,b),(c,c-a,a)|`
[on taking `(b-a)` common from `C_(1)` and `C_(3)` each]
`=(b-a)^(2)|(b-c,b-c,c),(a-b,a-b,b),(c-a,c-a,a)|[ :' C_(1)toC_(1)-C_(3)]`
`=0`
[since two columns `C_(1)` and `C_(2)` are identical so the value of determinant is zero]