Question

The value of the determinant $\left|\begin{array}{ccc}b+c& a-b& a\\ c+a& b-c& b\\ a+b& c-a& c\end{array}\right|$ , is

• A. $a^3+b^3+c^3-3abc$
• B. $3abc-a^3-b^3-c^3$
• C. $3abc+a^3+b^3+c^3$
• D. none of these

Answer 1

The correct option is B $3abc-a^3-b^3-c^3$

$\left|\begin{array}{ccc}b+c& a-b& a\\ c+a& b-c& b\\ a+b& c-a& c\end{array}\right|$
$C_1\to C_1+C_2+C_3$
$\left|\begin{array}{ccc}2(a+b+c)& 0& a+b+c\\ c+a& b-c& b\\ a+b& c-a& c\end{array}\right|$
$=(a+b+c)\left|\begin{array}{ccc}2& 0& 1\\ c+a& b-c& b\\ a+b& c-a& c\end{array}\right|$
$C_1\to C_1-{2C}_3$
$=(a+b+c)\left|\begin{array}{ccc}0& 0& 1\\ c+a-2b& b-c& b\\ a+b-2c& c-a& c\end{array}\right|$
$=(a+b+c)\left[\right(c-a\left)\right(c+a-2b)-(b-c\left)\right(a+b-2c\left)\right]=(a+b+c)[{-a}^2-b^2{-c}^2+ab+bc+ac]$
$=3abc-a^3{-b}^3{-c}^3$