<p>Let <em>A</em> be a square matrix such that \(A(\text{adj. }A) = \begin{bmatrix}4 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 4\end{bmatrix}\). Find the value of \(|\text{adj. }(\text{adj. }A)|\).</p>
Step-by-Step Solution
Key Concept: Use the fundamental property A(adj A) = |A|I to find |A|, then apply the formula |adj(adj A)| = |A|^(n-1)² for an n×n matrix.
<p><strong>Step 1:</strong> Use the property A(adj A) = |A|I</p><p>From the given: A(adj A) = 4I</p><p>Comparing with A(adj A) = |A|I, we get |A| = 4</p><p><strong>Step 2:</strong> Find |adj A| using the formula |adj A| = |A|^(n-1)</p><p>For a 3×3 matrix: |adj A| = |A|^(3-1) = |A|² = 4² = 16</p><p><strong>Step 3:</strong> Find |adj(adj A)| using the same formula applied to adj A</p><p>|adj(adj A)| = |adj A|^(n-1) = |adj A|² = 16² = 256</p><p><strong>Alternatively:</strong> Use the direct formula |adj(adj A)| = |A|^((n-1)²)</p><p>|adj(adj A)| = 4^((3-1)²) = 4^4 = 256</p><p>∴ Answer: <strong>256</strong></p>
Correct Answer: 256