Matrices & Determinants
Adjoint of a Matrix
Grade 12
Question:
<p>Let <em>A</em> be a 2 × 2 matrix.<br><strong>Statement 1:</strong> adj(adj <em>A</em>) = <em>A</em><br><strong>Statement 2:</strong> |adj <em>A</em>| = |<em>A</em>|</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 true, statement 2 is false.</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, but Statement 2 is always true for 2×2 matrices.
<p><strong>Step 1: Verify Statement 1 (adj(adj A) = A)</strong></p><p>For an n×n matrix: adj(adj A) = |A|^(n-2) · A</p><p>For n=2: adj(adj A) = |A|^(2-2) · A = |A|^0 · A = A ✓</p><p>Statement 1 is <strong>TRUE</strong></p><p><strong>Step 2: Verify Statement 2 (|adj A| = |A|)</strong></p><p>For an n×n matrix: |adj A| = |A|^(n-1)</p><p>For n=2: |adj A| = |A|^(2-1) = |A| ✓</p><p>Statement 2 is <strong>TRUE</strong></p><p><strong>Step 3: Determine relationship</strong></p><p>Both statements are correct, and Statement 2 follows from the general determinant property of adjugate matrices.</p><p>∴ Answer: B (Both statements are true, Statement 2 explains Statement 1)</p>
Correct Answer: B