Question

If p + q + r = a + b + c = 0, then the determinant $\mathrm{\Delta }=\left|\begin{array}{lll}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$ equals

• (A) 0
• (B) 1
• (C) pa + qb + rc
• (D) None of these

Answer 1

The Correct Option is A

Explanation:
We have $\mathrm{\Delta }=pqr(a^3+b^3+c^3)-abc(p^3+q^3+r^3)$But $a+b+c=0\Rightarrow (a+b{)}^3=-c^3$$\Rightarrow a^3+b^3+3ab(a+b)+c^3=0$$\Rightarrow a^3+b^3+c^3=-3ab(-c)=3abc$Similarly, $p^3+q^3+r^3=3pqr$Thus, $\mathrm{\Delta }=pqr(3abc)-abc(3pqr)=0$