1. The problem states that the determinant of the 3x3 matrix $$\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix} = (a - b)(b - c)(c - a).$$ 2. This matrix is a Vandermonde matrix for variables $a$, $b$, and $c$. 3. Recall that the determinant of a Vandermonde matrix for three variables is given by the product of the differences between each pair: $$\det = (a - b)(b - c)(c - a).$$ 4. To understand why, consider expanding the determinant using the standard cofactor expansion or by row operations to get an upper-triangular form. The differences arise because of the structure of the matrix where each row corresponds to powers of each variable. 5. This confirms the identity as stated. Therefore, the determinant indeed equals $$(a - b)(b - c)(c - a).$$