Question

Prove that :
$\left|\begin{array}{ccc}-a^2& ab& ac\\ ab& -b^2& bc\\ ac& bc& -c^2\end{array}\right|=4a^2{b}^2{c}^2$

Answer 1

$f\left(x\right)=\frac{x}{x+2}=y{f}^{\perp }\left(x\right)=\frac{2x}{1-x}$

$x=yx+2y$

$x(1-y)=2y$

$x=\frac{2y}{1-y}$

$\left|\begin{array}{ccc}-a^2& ab& ac\\ ab& -b^2& bc\\ ac& bc& -c^2\end{array}\right|=a^2\left|\begin{array}{ccc}1& b& c\\ ab& -b^2& bc\\ c& bc& -c^2\end{array}\right|$

$a^2{b}^2{c}^2\left|\begin{array}{ccc}-1& 1& 1\\ 1& -1& 1\\ 1& 1& -1\end{array}\right|=a^2{b}^2{c}^2\left(4\right)=4a^2{b}^2{c}^2$