Question

Find the value of the determinant \(\begin{vmatrix}a&b&c\\a^2&b^2&c^2\\a^3&b^3&c^3\end{vmatrix}\).

Hint:

Factor a, b, c out of the columns to expose a Vandermonde determinant, then apply its known factorization.

Answer 1

Step 1: Factor each column:
Column 1 is \(a,a^2,a^3=a(1,a,a^2)\); similarly column 2 factors out \(b\), column 3 factors out \(c\). Pulling these common factors out of each column:
\[ \begin{vmatrix}a&b&c\\a^2&b^2&c^2\\a^3&b^3&c^3\end{vmatrix}=abc\begin{vmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{vmatrix} \]

Step 2: Recognize the remaining determinant as a Vandermonde determinant:
\[ \begin{vmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{vmatrix}=(b-a)(c-a)(c-b) \]

Step 3: Rewrite the sign in the conventional \((a-b)(b-c)(c-a)\) form:
\((b-a)(c-a)(c-b)=(a-b)(b-c)(c-a)\) (each pair of sign flips cancels — two flips leave the product unchanged).

Final Answer:
\[ \boxed{abc(a-b)(b-c)(c-a)} \]