<p>If \(p + q + r = 0 = a + b + c\), then the value of the determinant \(\begin{vmatrix} pa & qb & rc \\ qc & ra & pb \\ rb & pc & qa \end{vmatrix}\) is</p>
Step-by-Step Solution
Key Concept: When p + q + r = 0 and a + b + c = 0, the determinant has a special structure where each row/column sum equals zero, making one row/column a linear combination of others. Use the constraint to express one variable in terms of others (e.g., r = -(p+q)) and simplify.
<p><strong>Step 1:</strong> Add all three rows of the determinant:</p><p>Row₁ + Row₂ + Row₃ = [(pa+qc+rb), (qb+ra+pc), (rc+pb+qa)]</p><p><strong>Step 2:</strong> Rearrange each element using commutativity:</p><p>= [(pa+qc+rb), (qb+ra+pc), (rc+pb+qa)]</p><p>= [p(a+b+c) + q(c+a+b) + r(b+c+a), ...]</p><p>= [(p+q+r)(a+b+c), (p+q+r)(a+b+c), (p+q+r)(a+b+c)]</p><p><strong>Step 3:</strong> Since p + q + r = 0 and a + b + c = 0, the sum becomes:</p><p>Row₁ + Row₂ + Row₃ = [0, 0, 0]</p><p><strong>Step 4:</strong> When one row becomes all zeros (linear dependence), the rows are linearly dependent, so the determinant equals zero.</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: A