Question

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

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

Answer 1

The correct option is A $0$

$\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$
$=pa(a^{2}qr-p^{2}bc)-qb(q^{2}ac-pr{b}^2)+rc(pq{c}^2-r^{2}ba)$
$=a^{3}pqr-p^{3}abc-q^{3}abc+pqr{b}^3+pqr{c}^3-r^{3}abc$
$=pqr(a^3+b^3+c^3)-abc(p^3+q^3+r^3)$
$\because a+b+c=0=p+q+r\Rightarrow a^3+b^3+c^3=3abc,p^3+q^3+r^3=3pqr$
so, the above value becomes,
$△=3\left(pqr\right)\left(abc\right)-3\left(abc\right)\left(pqr\right)=0$