Matrices & Determinants
Adjoint and Inverse of a Matrix
Grade 12
Question:
<p>Let \(A\) be a \(2 \times 2\) matrix.<br><b>Statement-1:</b> adj(adj \(A\)) = \(A\)<br><b>Statement-2:</b> |adj \(A\)| = |\(A\)|</p>
<p>Statement-1 is true, Statement-2 is false.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.</p>
<p>Statement-1 is false, Statement-2 is true.</p>
Step-by-Step Solution
Key Concept: For a 2×2 matrix A: adj(adj A) = |A|·A (not just A), and |adj A| = |A|^(n-1) where n=2, so |adj A| = |A|. Statement-1 is true only when |A|=1, making Statement-2 the universally true property.
<p><strong>Step 1:</strong> Recall the fundamental property: A·adj(A) = |A|·I</p><p><strong>Step 2:</strong> Find adj(adj A) by applying the property twice:</p><p>adj(A)·adj(adj A) = |adj A|·I</p><p><strong>Step 3:</strong> For a 2×2 matrix, |adj A| = |A|^(2-1) = |A|</p><p>So: adj(A)·adj(adj A) = |A|·I</p><p><strong>Step 4:</strong> Multiply both sides by A on the left and use A·adj(A) = |A|·I:</p><p>A·adj(A)·adj(adj A) = A·|A|·I</p><p>|A|·I·adj(adj A) = |A|·A</p><p>adj(adj A) = A (when |A| ≠ 0)</p><p><strong>Step 5:</strong> Verify Statement-2: |adj A| = |A|^(n-1) = |A|^1 = |A| ✓</p><p><strong>Analysis:</strong> Statement-1 is true (for invertible A), Statement-2 is true. Both statements are correct, and Statement-2 is the fundamental reason why Statement-1 holds.</p><p>∴ Answer: C (Both statements are true, and Statement-2 explains Statement-1)</p>
Correct Answer: C