Question

Find the value of: $\left|\begin{array}{ccc}b+c& a+b& a\\ c+a& b+c& b\\ a+b& c+a& c\end{array}\right|$

• A. $a^3+b^3+c^3$
• B. $3abc$
• C. $a^3+b^3+c^3-3abc$
• D. $a^2+b^2+c^2-ab-bc-ca$

Answer 1

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

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

Performing $C_2\to C_2-C_3$
$A=\left|\begin{array}{ccc}b+c& b& a\\ c+a& c& b\\ a+b& a& c\end{array}\right|$
Performing $C_1\to C_1-C_2$
$A=\left|\begin{array}{ccc}c& b& a\\ a& c& b\\ b& a& c\end{array}\right|$
$A=c(c^2-ab)-b(ac-b^2)+a(a^2-bc)$
$A=c^3-abc-bac+b^3+a^3-abc$

$A=a^3+b^3+c^3-3abc$