Question

If $\left|\begin{array}{ccc}a+b+2c& c& c\\ a& b+c+2a& a\\ b& b& c+a+2b\end{array}\right|=A(a+b+c{)}^3$, find the value of A.

Answer 1

Let $a+b+c=x$

$\left|\begin{array}{ccc}a+b+2c& c& c\\ a& b+c+2a& a\\ b& b& c+a+2b\end{array}\right|$

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

$=(x+c)\left[\right(x+a\left)\right(x+b)-ab]-c\left[a\right(x+b)-ab]+c[ab-b(x+a\left)\right]$

$=(x+c)(x^2+ax+bx)-acx-bcx$

$=x^3+a{x}^2+b{x}^2+c{x}^2+acx+bcx-acx-bcx$

$=x^3+x^2(a+b+c)$

$=x^3+x^3$

$=2x^3$

$\therefore A=2$