If a+b+c=4 and ab+bc+ca=0, then what is the value of the following determinant? vmatrix a & b & c b & c & a c & a & b vmatrix

NDA 2021 Mathematics question

If and , then what is the value of the following determinant?

(A)32
(B)Correct
(C)
(D)64

Answer: (B)

Solution

Expanding along the first row: a(bc-a^2)-b(b^2-ac)+c(ab-c^2)=3abc-(a^3+b^3+c^3). Using the identity a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca), the determinant equals

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Question ID VK-033496