<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 values of:</p><p>(i) |adj. A|</p>
Step-by-Step Solution
Key Concept: Use the fundamental property A(adj A) = |A|I to extract |A|, then apply |adj A| = |A|^(n-1) for an n×n matrix to find the determinant of the adjugate.
<p><strong>Step 1:</strong> Use the fundamental property of adjugate matrices: <br/>A(adj A) = |A|·I, where I is the identity matrix.</p><p><strong>Step 2:</strong> From the given equation A(adj A) = 4I₃, we have:<br/>|A|·I₃ = 4I₃<br/>Therefore, |A| = 4</p><p><strong>Step 3:</strong> Apply the determinant property for adjugate matrices:<br/>For an n×n matrix, |adj A| = |A|^(n-1)<br/>Here n = 3, so: |adj A| = |A|^(3-1) = |A|²</p><p><strong>Step 4:</strong> Substitute |A| = 4:<br/>|adj A| = 4² = 16</p><p><strong>Verification:</strong> Taking determinant of both sides of A(adj A) = 4I₃:<br/>|A|·|adj A| = |4I₃| = 4³ = 64<br/>4·|adj A| = 64 ⟹ |adj A| = 16 ✓</p><p>∴ Answer: <strong>|adj A| = 16</strong></p>
Correct Answer: 16