Determinants: PYQ Walkthrough (6)

Question

Without expanding, prove that

$\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix} = (a-b)(b-c)(c-a)$

This is the Vandermonde determinant — appears every year in JEE/CBSE.

Solution — Step by Step

Final answer: $(a-b)(b-c)(c-a)$. Proved.

Why This Works

Row operations don’t change the determinant value (as long as we don’t multiply a row by a constant — that scales the determinant). Subtraction creates zeros, which makes cofactor expansion almost trivial.

The Vandermonde structure repeats in higher orders too: a $4\times 4$ Vandermonde gives the product of all pairwise differences of the variables. The $3\times 3$ case is what JEE/CBSE tests.

Alternative Method

Expand directly: $1(bc^2 - cb^2) - a(c^2 - b^2) + a^2(c - b)$. Group terms and factor. Takes about 3 minutes vs 1 minute via row ops — the row operation method dominates.

Common Mistake

Multiplying a row by $-1$ to get $a^2 - b^2$ instead of $b^2 - a^2$ without flipping the sign of the determinant. Row operations $R_i \to R_i + kR_j$ are sign-preserving, but $R_i \to -R_i$ flips the sign.