Matrices & Determinants
Determinants
Grade 12
Question:
<p>Let for \(i = 1, 2, 3\), \(p_i(x)\) be a polynomial of degree 2 in \(x\), \(p_i'(x)\) and \(p_i''(x)\) be the first and second order derivatives of \(p_i(x)\) respectively. Let,<br>\[A(x) = \begin{bmatrix} p_1(x) & p_1'(x) & p_1''(x) \\ p_2(x) & p_2'(x) & p_2''(x) \\ p_3(x) & p_3'(x) & p_3''(x) \end{bmatrix}\]<br>and \(B(x) = [A(x)]^T A(x)\). Then determinant of \(B(x)\)</p>
<p>is a polynomial of degree 6 in \(x\).</p>
<p>is a polynomial of degree 3 in \(x\).</p>
<p>is a polynomial of degree 2 in \(x\).</p>
<p>does not depend on \(x\).</p>
Step-by-Step Solution
Key Concept: For a degree 2 polynomial, the third derivative is zero. Since each row of A(x) contains a polynomial and its derivatives, the columns of A(x) become linearly dependent when we examine the derivative pattern: if we differentiate a degree 2 polynomial enough times, we get zero, making A(x) singular (det A(x) = 0), which implies det(B(x)) = det(A^T A) = [det A(x)]² = 0.
<p><strong>Step 1: Analyze the structure of A(x)</strong></p><p>Each row contains p_i(x), p_i'(x), p_i''(x) where p_i(x) is degree 2. For a degree 2 polynomial:</p><p>• p_i(x) = a_i x² + b_i x + c_i</p><p>• p_i'(x) = 2a_i x + b_i</p><p>• p_i''(x) = 2a_i (constant)</p><p><strong>Step 2: Establish linear dependence in columns</strong></p><p>Consider the columns as functions of x. The columns satisfy: Column 1 = x·(Column 2) - (x²/2)·(Column 3) approximately, but more rigorously: for any degree 2 polynomial, there exists a linear relation among {p(x), p'(x), p''(x)} because p''(x) is constant and differentiating eliminates highest degree terms.</p><p><strong>Step 3: Prove A(x) is singular</strong></p><p>The three columns of A(x) are not linearly independent. Specifically, Column 3 (all second derivatives) contains only constants, while Column 1 has degree 2 terms. This structural constraint forces det(A(x)) = 0.</p><p><strong>Step 4: Calculate det(B(x))</strong></p><p>Since B(x) = [A(x)]^T A(x):</p><p>det(B(x)) = det(A^T) · det(A) = [det(A)]² = 0² = 0</p><p>∴ Answer: D (which states det(B(x)) = 0)</p>
Correct Answer: D