Matrices & Determinants
Idempotent matrix
Grade 12
Question:
<p>If \(A\), \(B\), \(A+I\), \(A+B\) are idempotent matrices, then \(AB\) is equal to</p>
<p>\(BA\)</p>
<p>\(-BA\)</p>
<p>\(I\)</p>
<p>\(O\)</p>
Step-by-Step Solution
Key Concept: An idempotent matrix M satisfies M² = M. Use this property on A, B, A+I, and A+B simultaneously to derive constraints on their products. The key is that idempotence forces AB = BA = 0.
<p><strong>Step 1:</strong> Write idempotency conditions for each matrix:</p><p>• A² = A</p><p>• B² = B</p><p>• (A+I)² = A+I</p><p>• (A+B)² = A+B</p><p><strong>Step 2:</strong> Expand (A+I)² = A+I:</p><p>A² + A + A + I = A + I</p><p>A + 2A + I = A + I (using A² = A)</p><p>2A = 0 ⟹ A = 0</p><p><strong>Step 3:</strong> Since A = 0, expand (A+B)² = A+B:</p><p>B² + B + B + 0 = 0 + B</p><p>B + 2B = B (using B² = B)</p><p>2B = 0 ⟹ B = 0</p><p><strong>Step 4:</strong> Calculate AB:</p><p>AB = 0 · 0 = 0 (null matrix)</p><p>∴ Answer: D</p>
Correct Answer: D