Question

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

Answer 1

Take out $a$, $b$, $c$ from $C_1$, $C_2$ and $C_3$ respectively.
$△=abc\left|\begin{array}{ccc}a& c& a+c\\ a+b& b& a\\ b& b+c& c\end{array}\right|$
Apply $C_1+(C_2-C_3)$ and take $2b$ common from $C_1$
$△=abc.2b\left|\begin{array}{ccc}0& c& a+c\\ 1& b& a\\ 1& b+c& c\end{array}\right|$
Apply $R_2-R_3$
$△={2ab}^{2}c\left|\begin{array}{ccc}0& c& a+c\\ 0& -c& a-c\\ 1& b+c& c\end{array}\right|$
${=2ab}^{2}c\{c(a-c)+c(a+c)\}$
${=2ab}^{2}c\left(2ac\right)=4a^2{b}^2{c}^2$