<s>[INST] Use row reduction to show that the determinant is equal to this variable. Show
determinant of:
\begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix}
is equal to $(b - a)(c - a)(c - b)$
I'm not sure if you can use squares or square roots hmmm.. please help me. I'm sure it's a simple question. Much appreciated.
 [/INST] Subtracting $\DeclareMathOperator{Col}{Col}\Col_1$ from $\Col_2$ and $\Col_3$ gives
$$
\begin{pmatrix}
1 & 0 & 0 \\
a & b-a & c-a \\
a^2 & b^2-a^2 & c^2-a^2 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 & 0 \\
a & b-a & c-a \\
a^2 & (b+a)(b-a) & (c+a)(c-a)
\end{pmatrix}
$$
The determinant is then
\begin{align*}
(b-a)(c+a)(c-a)-(c-a)(b+a)(b-a)
&=(b-a)(c-a)(c+a-b-a) \\
&=(b-a)(c-a)(c-b)
\end{align*}
Of course, if you'd rather use row operations then you could consider the transpose and use the fact that $\det(A^\top)=\det(A)$.
</s>