Matrices & Determinants
Determinants and algebraic identities
Grade 12
Question:
<p><strong>507.</strong> If \(a^2 + 8b^2 + 2c^2 + 2d^2 - 4ab - 4bc - 4bd = 0\) (where \(a, b, c, d \in \mathbb{R}\)), then the value of \(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\) is:</p>
<p>\(\dfrac{a^2}{4}\)</p>
<p>\(b^2\)</p>
<p>\(c^2\)</p>
<p>\(d^2\)</p>
Step-by-Step Solution
Key Concept: Rewrite the quadratic form as a sum of perfect squares to find constraints on variables, then use those constraints to evaluate the determinant. The expression must equal a sum of squares (SOS) decomposition.
<p><strong>Step 1:</strong> Rearrange the given equation as a sum of squares:</p><p>a² + 8b² + 2c² + 2d² - 4ab - 4bc - 4bd = 0</p><p><strong>Step 2:</strong> Group and complete squares strategically:</p><p>(a² - 4ab + 4b²) + 4b² + 2c² + 2d² - 4bc - 4bd = 0</p><p>(a - 2b)² + 4b² + 2c² + 2d² - 4bc - 4bd = 0</p><p><strong>Step 3:</strong> Continue completing squares with remaining terms:</p><p>(a - 2b)² + 4(b² - bc - bd) + 2c² + 2d² = 0</p><p>(a - 2b)² + 4(b² - bc - bd + c²/4 + d²/4 - c²/4 - d²/4) + 2c² + 2d² = 0</p><p>(a - 2b)² + 4(b - c/2 - d/2)² - c² - d² + 2c² + 2d² = 0</p><p>(a - 2b)² + 4(b - c/2 - d/2)² + c² + d² = 0</p><p><strong>Step 4:</strong> Since this is a sum of squares equaling zero, each square must be zero:</p><p>a - 2b = 0 ⟹ a = 2b</p><p>b - c/2 - d/2 = 0 ⟹ b = (c + d)/2</p><p>c² = 0 ⟹ c = 0</p><p>d² = 0 ⟹ d = 0</p><p><strong>Step 5:</strong> From c = 0 and d = 0: b = 0, and from a = 2b: a = 0</p><p><strong>Step 6:</strong> Calculate the determinant:</p><p>∣a b∣ = ∣0 0∣ = 0·0 - 0·0 = <strong>0</strong></p><p>∣c d∣ ∣0 0∣</p><p>∴ Answer: B (determinant = 0)</p>
Correct Answer: B