<p>If <em>A</em> is order 3 square matrix such that \(|A| = 2\), then \(|\text{adj}(\text{adj}(\text{adj } A))|\) is</p>
Step-by-Step Solution
Key Concept: Use the property that for an n×n matrix: |adj(A)| = |A|^(n-1). Apply this formula recursively three times, and recognize that adj(adj(A)) relates back to A through the identity adj(adj(A)) = |A|^(n-2)·A for n≥3.
<p><strong>Step 1:</strong> Recall the fundamental property: For an n×n matrix A, |adj(A)| = |A|^(n-1)</p><p><strong>Step 2:</strong> Find |adj(A)| where |A| = 2 and n = 3:<br/>|adj(A)| = |A|^(3-1) = 2^2 = 4</p><p><strong>Step 3:</strong> Find |adj(adj(A))| using |adj(A)| = 4 and n = 3:<br/>|adj(adj(A))| = |adj(A)|^(3-1) = 4^2 = 16</p><p><strong>Step 4:</strong> Find |adj(adj(adj(A)))| using |adj(adj(A))| = 16 and n = 3:<br/>|adj(adj(adj(A)))| = |adj(adj(A))|^(3-1) = 16^2 = 256</p><p><strong>Verification:</strong> Pattern shows: 2 → 2² = 4 → 4² = 16 → 16² = 256, which equals 2^8</p><p>∴ Answer: <strong>256</strong> (or equivalently 2^8)</p>
Correct Answer: A