First slide

Introduction to Determinants

Question

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

Moderate

A

0
B

pa + qb + rc
C

1
D

none of these

Solution

$|\begin{array}{lll} pa qb rc \\ qc ra pb \\ rb pc qa \end{array}|$
$\begin{array}{l} = pqr (a^{3} + b^{3} + c^{3} - 3 abc) - abc (p^{3} + q^{3} + r^{3} - 3 pqr) \\ = pqr (a + b + c) (a^{2} + b^{2} + c^{2} - ab - bc - ca) - abc (p + q + r) (p^{2} + q^{2} + r^{2} - pq - qr - pr) \\ = 0 \end{array}$

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