Matrices & Determinants
Determinant as a function
Grade 12
Question:
<p>Let \(A = [a_{ij}]\) be a matrix of order 3 where \[a_{ij} = \begin{cases} 0, & i = j \\ (i + 2j - 3)x, & i > j \\ i, & i < j \end{cases}.\] If \(f(x) = \det.(A)\), then which of the following is(are) correct statement(s)?</p>
<p>(a) \(\displaystyle\int_{-1}^{1} f(x)\,dx = \frac{8}{3}\)</p>
<p>(b) \(|f(|x|)|\) is non-differentiable at 2 points</p>
<p>(c) \(f(|x|) = k\) has four distinct solutions for \(k \in \left(0, \dfrac{1}{4}\right)\)</p>
<p>(d) \(\displaystyle\int_0^1 \frac{dx}{f(x) + 2x + 1} = \frac{1}{3}\)</p>
Step-by-Step Solution
Key Concept: Construct the matrix by carefully applying the piecewise definition to each position based on row-column relationships, then evaluate determinant properties and matrix operations systematically.
<p><strong>Step 1: Construct Matrix A</strong></p><p>Using the piecewise definition:</p><p>• For i=j: a<sub>ij</sub> = 0</p><p>• For i>j (lower triangle): a<sub>ij</sub> = (i+2j-3)x</p><p>• For i<j (upper triangle): a<sub>ij</sub> = i</p><p>$$A = \begin{pmatrix} 0 & 1 & 1 \\ (3x) & 0 & 2 \\ (7x) & (9x) & 0 \end{pmatrix}$$</p><p><strong>Step 2: Evaluate |A|</strong></p><p>A is a lower triangular matrix with all diagonal elements = 0. Therefore:</p><p>|A| = 0 · 0 · 0 = 0 (product of diagonal elements)</p><p>This means A is singular (non-invertible).</p><p><strong>Step 3: Check Each Statement</strong></p><p><strong>Option A:</strong> |A| = 0 ✓ TRUE</p><p><strong>Option B:</strong> A is invertible – FALSE (since |A| = 0)</p><p><strong>Option C:</strong> A is lower triangular – TRUE ✓ (zeros above diagonal with (i+2j-3)x and 0 on diagonal)</p><p><strong>Option D:</strong> Trace(A) = 0 – TRUE ✓ (sum of diagonal = 0+0+0 = 0)</p><p>∴ Answer: <strong>ACD</strong></p>
Correct Answer: ACD