Question

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

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

Answer 1

Answer: (A)

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