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. $pq+qb+ra$
• C. $1$
• D. none of these

Answer 1

The correct option is A $0$

Given , $p+q+r=0=a+b+c$
$\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$
$=pa(qr{a}^2-p^{2}bc)-qb(q^{2}ac-pr{b}^2)+rc(pq{c}^2-r^{2}ab)$
$=pqr{a}^3-p^{3}abc-q^{3}abc+pqr{b}^3+pqr{c}^3-r^{3}abc$
$=pqr(a^3+b^3+c^3)-abc(p^3+q^3+r^3)$
$=pqr[{(a+b+c)}^3+3abc(a+b+c\left)\right]-abc[{(p+q+r)}^3+3pqr(p+q+r\left)\right]$
$=0$