1. ## matrix

Show that $\displaystyle \det \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{bmatrix} = (b-a)(c-a)(c-b)$.

Using expansion by minors I got $\displaystyle c^{2}(b-a) + b^{2}(a-c) + a^{2}(c-b)$. How would I convert this to the above form?

2. I think I got it.

Thanks

3. Originally Posted by heathrowjohnny
Show that $\displaystyle \det \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{bmatrix} = (b-a)(c-a)(c-b)$.

Using expansion by minors I got $\displaystyle c^{2}(b-a) + b^{2}(a-c) + a^{2}(c-b)$. How would I convert this to the above form?
$\displaystyle = c^{2}(b-a) + b^2 a - b^2 c + a^2 c- a^2 b = c^{2}(b-a) + b^2 a - a^2 b - b^2 c + a^2 c$

$\displaystyle = c^2 (b - a) + ab (b - a) - c (b^2 - a^2)$

$\displaystyle = c^2 (b - a) + ab (b - a) - c (b - a)(b + a) = (b - a)(c^2 + ab - cb - ca)$

$\displaystyle = (b - a)(c^2 - cb + ab - ca) = (b - a)(c[c - b] - a[c - b]) = (b - a)(c - a)(c - b)$.

Edit: Well, for what it's worth .....