Question

Prove that:
$\left|\begin{array}{ccc}a& c& a+c\\ a+b& b& a\\ b& b+c& c\end{array}\right|=4abc$

Answer 1

To prove: $\left|\begin{array}{ccc}a& c& a+c\\ a+b& b& a\\ b& b+c& c\end{array}\right|=4abc$

Lets take LHS $=\left|\begin{array}{ccc}a& c& a+c\\ a+b& b& a\\ b& b+c& c\end{array}\right|$

$=a[bc-a(b+c\left)\right]-c\left[c\right(a+b)-ab]+(a+c)\left[\right(a+b\left)\right(b+c)-b^2]$

$=a(bc-ab-ac)-c(ac+bc-ab)+(a+c)(ab+ac+b^2+bc-b^2)$

$=a(bc-ab-ac)-c(ac+bc-ab)+(a+c)(ab+ac+bc)$

$=abc-a^{2}b-a^{2}c-a{c}^2-b{c}^2+abc+a^{2}b+a^{2}c+abc+abc+a{c}^2+b{c}^2$

$=abc+abc+abc+abc$

Therefore, $\left|\begin{array}{ccc}a& c& a+c\\ a+b& b& a\\ b& b+c& c\end{array}\right|=4abc$