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Question: The value of the determinant $\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix...

The value of the determinant ∣111abca2b2c2∣=\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix} =

A

(a-b)(b-c)(c-a)(a-b-c)

B

(a-b)(b-c)(c-a)(a+b+c)

C

(a-b)(b-c)(c-a)

D

(a+b)(b+c)(c+a)(a+b+c)

Answer

(a-b)(b-c)(c-a)

Explanation

Solution

The given determinant is a Vandermonde determinant. It can be evaluated using elementary column operations.

Steps:

  1. Apply column operations to create zeros in the first row.

  2. Expand the determinant along the first row.

  3. Factor out common terms.

  4. Evaluate the remaining 2x2 determinant.

  5. Rearrange the terms to match the options.

The determinant simplifies to (a−b)(b−c)(c−a)(a-b)(b-c)(c-a).