Question

If $\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ {(a+1)}^2& {(b+1)}^2& {(c+1)}^2\\ {(a-1)}^2& {(b-1)}^2& {(c-1)}^2\end{array}\right|=k(a-b)(b-c)(c-a)$, then find the value of $-k$

Answer 1

$\Delta =\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ {(a+1)}^2& {(b+1)}^2& {(c+1)}^2\\ {(a-1)}^2& {(b-1)}^2& {(c-1)}^2\end{array}\right|$
$=\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ a^2+2a+1& b^2+2b+1& c^2+2c+1\\ a^2-2a+1& b^2-2b+1& c^2-2c+1\end{array}\right|$
$=\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ 2a+1& 2b+1& 2c+1\\ -4a& -4b& -4c\end{array}\right|$ $R_2\to R_2-R_1,$ and $R_3\to R_3-R_2$
$=\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ 4a& 4b& 4c\\ 2a+1& 2b+1& 2c+1\end{array}\right|=4\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ a& b& c\\ 2a+1& 2b+1& 2c+1\end{array}\right|$
$=4\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ a& b& c\\ 1& 1& 1\end{array}\right|$ $R_3\to R_2-2R_2$
$=-4\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|=-4(a-b)(b-c)(c-a)$