Consider the following determinant: delta = |1 a bc| |1 b ca| |1 c ab| Which of the following is a factor of delta? A. a + b B. a - b C. a + b + c D. abc

GATE 1998 · Engineering Mathematics · Matrix · medium

Answer: B. (a - b) is a factor. The full factorization is delta = (a-b)(b-c)(c-a).

1. Apply R1 -> R1 - R2 and R2 -> R2 - R3: R1 - R2: [1-1, a-b, bc-ca] = [0, a-b, c(b-a)] = [0, (a-b), -c(a-b)]. R2 - R3: [0, b-c, ca-ab] = [0, (b-c), -a(b-c)]. R3 stays: [1, c, ab].
2. Factor out from rows: Factor (a-b) from row 1 and (b-c) from row 2. The remaining determinant: expand along column 1: only row 3 contributes = 1 * |1 -c; 1 -a| = 1*(-a) - 1*(-c) = c - a = -(a-c).
3. Identify the factor from options: delta = (a-b)(b-c)(c-a). Among the options: A. a+b — not a factor. B. a-b — YES, this is a factor. C. a+b+c — not a factor. D. abc — not a factor.