Matrices & Determinants
Determinant Bounds and Optimization
Grade 12
Question:
<p>Find the maximum value of the determinant of an arbitrary 3 × 3 matrix A, each of whose entries \(a_{ij} \in \{-1, 1\}\).</p>
Step-by-Step Solution
Key Concept: Test various ±1 matrices systematically; the maximum determinant occurs at specific orthogonal-like configurations
<p><strong>Solution:</strong> We need to find the maximum determinant when all entries are ±1.</p><p>For a 3 × 3 matrix with entries in {−1, 1}, the determinant can be computed by checking various configurations.</p><p>Consider the matrix: $$A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & -1 \\ 1 & -1 & 1 \end{pmatrix}$$</p><p>Computing the determinant: $\det(A) = 1(1 + 1) - 1(1 + 1) + 1(-1 - 1) = 2 - 2 - 2 = -2$</p><p>Through systematic search, the maximum value is found to be <strong>4</strong>, achieved by matrices such as certain Hadamard-like configurations.</p><p>∴ Maximum det(A) = <strong>4</strong></p>
Correct Answer: 4