<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] We see immmediatley that the determinant is a polynomial in $a,b,c$, homgenous of degree $3$. And if $a=b$ or $b=c$ or $c=a$, two columns are identical and make the determiniant zero.
On the other hand, if $a,b,c$ are pairwise distinct, the determinant must be nonzero as any vanishing linear combination of rows would result in a quadratic polynomial having three roots $a,b,c$.
If these arguments are not convincing enough, just employ Gauss's algorithm:
$$\begin{align}\det\begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix} 
&=\det\begin{pmatrix}1&1&1\\0&b-a&c-a\\0&b^2-a^2&c^2-a^2\end{pmatrix}\\ 
&=\det\begin{pmatrix}1&1&1\\0&b-a&c-a\\0&0&c^2-a^2-(b+a)(c-a)\end{pmatrix}\\
&=(b-a)\cdot(c^2-a^2-(b+a)(c-a))\\
&=(b-a)(c-a)(c+a-(b+a))\\&=(b-a)(c-a)(c-b) \end{align}
$$
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