Matrices & Determinants
Determinant Equations
Grade 12
Question:
<p>If \(f(x) = \begin{vmatrix} x & a & a \\ a & x & a \\ a & a & x \end{vmatrix} = 0\), then</p>
<p>(1) \(f'(x) = 0\) and \(f''(x) = 0\) has one common root</p>
<p>(2) \(f(x) = 0\) and \(f'(x) = 0\) has one common root</p>
<p>(3) sum of roots of \(f(x) = 0\) is \(-3a\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: Factor the determinant by recognizing that adding all rows creates a common factor (x + 2a), then reduce to a 2×2 determinant. The roots occur when either (x + 2a) = 0 or the reduced determinant equals zero.
<p><strong>Step 1:</strong> Add all three rows together: R₁ + R₂ + R₃ gives a row where each entry is (x + 2a).</p><p><strong>Step 2:</strong> Factor out (x + 2a): </p><p>f(x) = (x + 2a)∣1 & a & a \\ a & x & a \\ a & a & x∣</p><p><strong>Step 3:</strong> Subtract the first row from rows 2 and 3 in the remaining determinant:</p><p>f(x) = (x + 2a)∣1 & a & a \\ a-1 & x-a & 0 \\ a-1 & 0 & x-a∣</p><p><strong>Step 4:</strong> Expand along the third column: the 2×2 minor gives (x - a)²</p><p><strong>Step 5:</strong> f(x) = (x + 2a)(x - a)² = 0</p><p>∴ <strong>Roots are x = -2a (single root) and x = a (double root)</strong></p>
Correct Answer: B