The value of the determinant of the matrix $\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|$ is:

Solution

1. Let the determinant be $D=\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|$.
2. Perform row operations $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$ to obtain $D=\left|\begin{array}{ccc}1& a& a^2\\ 0& b-a& b^2-a^2\\ 0& c-a& c^2-a^2\end{array}\right|$.
3. Factor out $(b-a)$ from the second row and $(c-a)$ from the third row to get $D=(b-a)(c-a)\left|\begin{array}{ccc}1& a& a^2\\ 0& 1& b+a\\ 0& 1& c+a\end{array}\right|$.
4. Expand the determinant along the first column to get $$\begin{array}{rl}D& =(b-a)(c-a)((c+a)-(b+a))\\ & =(b-a)(c-a)(c-b)\end{array}$$.
5. Rearrange the factors to match the standard form $(a-b)(b-c)(c-a)$.
6. Hence the answer is (A).