Question

The value of 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. $-(a^3+b^3+c^3-3abc)$
• C. $-(a^3+b^3+c^3)$
• D. None of these

Answer 1

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

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

Applying $C_1\to C_1+C_3,C_2\to C_2-C_3,$ we get

$\Delta =\left|\begin{array}{ccc}a+b+c& -b& a\\ a+b+c& -c& b\\ a+b+c& -a& c\end{array}\right|=-(a+b+c)\left|\begin{array}{ccc}1& b& a\\ 1& c& b\\ 1& a& c\end{array}\right|$

Applying $R_2\to R_2-R_1,R_3\to R_3-R_1,$ we get

$\Delta =-(a+b+c)\left|\begin{array}{ccc}1& b& a\\ 0& c-b& b-a\\ 0& a-b& c-a\end{array}\right|=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=-(a^3+b^3+c^3-3abc)$