Question

If $p+q+r=0=a+b+c,$ then the determinant $△=\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$ is equal to

• A. $0$
• B. $1$
• C. $pa+qb+rc$
• D. None of these

Answer 1

The correct option is A $0$

Expanding along $R_1:$
$\Rightarrow △=pa(a^{2}qr-p^{2}bc)-qb(q^{2}ac-b^{2}pr)+rc(c^{2}pq-r^{2}ab)\Delta =pqr(a^3+b^3+c^3)-abc(p^3+q^3+r^3)\cdots \left(i\right)$
As, $p+q+r=0=a+b+c$
$\Rightarrow a^3+b^3+c^3=3abc$ and $p^3+q^3+r^3=3pqr$
So, $△=0$ [from $\left(i\right)$]