Question

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

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

Answer 1

The correct option is A $0$

$\Delta =pqr(a^3+b^3+c^3)-abc(p^3+q^3+r^3)$
But $a+b+c=0\Rightarrow {(a+b)}^3=-c^2$
$\Rightarrow a^3+b^3+3ab(a+b)+c^3=0$
$\Rightarrow a^3+b^3+c^3=-3ab(-c)=3abc$
Similarly $p^3+q^3+r^3=3pqr$
Thus $\Delta =pqr\left(3abc\right)-abc\left(3pqr\right)=0$