Question

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

Answer 1

$LHS=\left|\begin{array}{ccc}{a}^2& bc& ac+c^2\\ a^2+ab& b^2& ac\\ ab& b^2+bc& c^2\end{array}\right|=abc\left|\begin{array}{ccc}a& c& a+c\\ a+b& b& a\\ b& b+c& c\end{array}\right|$
(Take out a from $C_1,b$ from $C_2$ and c from $C_3$)
$=abc\left|\begin{array}{ccc}0& c& a+c\\ 2b& b& a\\ 2b& b+c& c\end{array}\right|$ (using $C_1\to C_1+C_2-C_3$)
$=abc\left|\begin{array}{ccc}0& c& a+c\\ 0& -c& a-c\\ 2b& b+c& c\end{array}\right|$
[using $R_2\to R_2-R_3$]
Expanding along $C_1$, we get $=\left(abc\right)\left[\right(2b\left)\left\{c\right(a-c)+c(a+c\left)\right\}\right]$
$=2\left(a{b}^{2}c\right)\left(2ac\right)=4a^2{b}^2{c}^2=RHS$.
Hence proved.