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: Pull the common factor from each column explicitly:
Column 1 entries are $a\cdot1,\ a\cdot a,\ a\cdot a^2$, so factoring $a$ leaves $1,a,a^2$ in that column; same logic for columns 2 and 3 with $b$ and $c$.

Step 2: The reduced matrix is the transpose-style Vandermonde in rows $1,x,x^2$:
$\begin{vmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{vmatrix}$ is a standard Vandermonde determinant with value $(b-a)(c-a)(c-b)$.

Step 3: Multiply back in the factored constants:
Total value $=abc\cdot(b-a)(c-a)(c-b)$.

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