<s>[INST] Proving determinant with variables I have a problem that asks:
Prove that det$\begin{pmatrix} 
1 && 1 && 1 \\ 
a && b && c \\
a^2 && b^2 && c^2 \end{pmatrix} = (c-a)(c-b)(b-a)$
I was thinking of solving it like normal and see if I can end up with $(c-a)(c-b)(b-a)$
What I did:
det$\begin{pmatrix}
b && c \\
b^2 && c^2 \end{pmatrix}$ - 
det$\begin{pmatrix}
a && c \\
a^2 && c^2 \end{pmatrix}$ +
det$\begin{pmatrix}
a && b \\
a^2 && b^2 \end{pmatrix}$
which gives 
$(bc^2 - cb^2)-(ac^2-ca^2)+(ab^2-ba^2)$
which simplifies to
$(b-a)c^2 + (a-c)b^2 + (c-b)a^2$ 
and this is where I got stuck. Am I even approaching this problem correctly? Any help would be appreciated.
 [/INST] If a=b, b=c, c=a then determinant value is 0 so
  (a-b)(b-c)(c-a) are factor of the determinant now comparing the leading coefficient so the Value of the determinant is (a-b)(b-c)(c-a) 
</s>