Matrices & Determinants
Adjoint and Inverse of a Matrix
Grade None
Question:
<p>If \(A\) is a \(3 \times 3\) matrix such that \(|5 \cdot \text{adj}\, A| = 5\), then \(|A|\) is equal to</p>
<p>\(\pm \dfrac{1}{5}\)</p>
<p>\(\pm 5\)</p>
<p>\(\pm 1\)</p>
<p>\(\pm \dfrac{1}{25}\)</p>
Step-by-Step Solution
Key Concept: Use the property that |adj A| = |A|^(n-1) for an n×n matrix, and |kB| = k^n|B| for an n×n matrix B. Here n=3, so |5·adj A| = 5³|adj A| = 125|A|².
<p><strong>Step 1:</strong> Apply the scalar multiplication property of determinants.</p><p>For a 3×3 matrix M: |5M| = 5³|M| = 125|M|</p><p><strong>Step 2:</strong> Apply to our problem: |5·adj A| = 125|adj A|</p><p><strong>Step 3:</strong> Use the fundamental property: |adj A| = |A|^(n-1) = |A|^(3-1) = |A|²</p><p><strong>Step 4:</strong> Set up the equation: 125|A|² = 5</p><p><strong>Step 5:</strong> Solve: |A|² = 5/125 = 1/25</p><p><strong>Step 6:</strong> Therefore: |A| = ±1/5</p><p>∴ Answer: A (assuming |A| = 1/5 or answer options include ±1/5)</p>
Correct Answer: A