<p>If <i>A</i> is unimodular, then which of the following is unimodular?</p>
<p>\(-A\)</p>
<p>\(A^{-1}\)</p>
<p>\(\text{adj}\, A\)</p>
<p>\(\omega A\), where \(\omega\) is cube root of unity</p>
Step-by-Step Solution
Key Concept: A matrix is unimodular if its determinant equals ±1. Use determinant properties: det(AB) = det(A)det(B), det(A^T) = det(A), and det(A^n) = [det(A)]^n to identify which operations preserve the unimodular property.
<p><strong>Definition:</strong> A matrix A is unimodular if det(A) = ±1.</p><p><strong>Step 1:</strong> If det(A) = ±1, then det(A^T) = det(A) = ±1, so <strong>A^T is unimodular</strong>.</p><p><strong>Step 2:</strong> Since det(A) = ±1 ≠ 0, A is invertible. Then det(A^(-1)) = 1/det(A) = ±1, so <strong>A^(-1) is unimodular</strong>.</p><p><strong>Step 3:</strong> For any positive integer n: det(A^n) = [det(A)]^n = (±1)^n = ±1, so <strong>A^n is unimodular</strong>.</p><p><strong>Step 4:</strong> For scalar multiplication: det(kA) = k^n·det(A) = k^n(±1). This equals ±1 only if k = ±1, so kA is NOT generally unimodular.</p><p>∴ Answer: A (A^T), B (A^(-1)), C (A^n)</p>
Correct Answer: A,B,C