<p>If \(B\) is an idempotent matrix, and \(A = I - B\), then</p>
Step-by-Step Solution
Key Concept: An idempotent matrix B satisfies B² = B. Use this property to find relationships between A and B, then compute A² and AB to verify algebraic identities.
<p><strong>Given:</strong> B is idempotent, so B² = B, and A = I - B</p><p><strong>Step 1:</strong> Find A²</p><p>A² = (I - B)² = I² - 2IB + B² = I - 2B + B</p><p>Since B² = B: A² = I - 2B + B = I - B = A</p><p>✓ <strong>A is idempotent</strong></p><p><strong>Step 2:</strong> Find AB</p><p>AB = (I - B)B = B - B² = B - B = 0</p><p>✓ <strong>AB = 0 (null matrix)</strong></p><p><strong>Step 3:</strong> Find BA</p><p>BA = B(I - B) = B - B² = B - B = 0</p><p>✓ <strong>BA = 0 (null matrix)</strong></p><p><strong>Step 4:</strong> Verify A + B = I</p><p>A + B = (I - B) + B = I</p><p>✓ <strong>A + B = I</strong></p><p><strong>Step 5:</strong> Check if A and B are orthogonal</p><p>Since AB = 0 and BA = 0, A and B are orthogonal (their product is zero)</p><p>∴ Answer: <strong>A, C, D</strong> (A² = A; AB = 0; A + B = I are the three correct statements)</p>
Correct Answer: A,C,D