Question

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

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

Answer 1

Answer: (A)

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