<p>If \(AB = A\) and \(BA = B\), then which of the following is/are true?</p>
<p>\(A\) is idempotent</p>
<p>\(B\) is idempotent</p>
<p>\(A^T\) is idempotent</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: If AB = A and BA = B, then B is a left inverse of A and A is a left inverse of B simultaneously. This forces A² = A and B² = B (both are idempotent), and combined with the given conditions, it implies A = B.
<p><strong>Step 1:</strong> From AB = A, multiply both sides by B on the right: AB² = AB = A</p><p><strong>Step 2:</strong> From BA = B, we have B² = BAB. But also BA = B gives us B(AB) = B(A) = BA = B, so B² = B (idempotent)</p><p><strong>Step 3:</strong> From AB = A, multiply by B on the left: B(AB) = BA, so BA = BA, which checks. From BA = B, multiply by A on the right: BA² = BA = B</p><p><strong>Step 4:</strong> From AB = A and BA = B: A = AB = A(BA) = (AB)A = AA = A². So A² = A (idempotent)</p><p><strong>Step 5:</strong> From A² = A and B² = B, and the condition AB = A: A(B - I) = 0. From BA = B: (A - I)B = 0. This means A and B have compatible structures. Multiplying AB = A by B from left: BAB = BA = B, and AB = A is already given. From BA = B: BAB = BB = B², but BAB = B and B² = B, consistent. The conditions force A = B.</p><p><strong>Step 6:</strong> Verify: If A = B, then AB = AA = A² = A ✓ and BA = AA = A² = A = B ✓. Therefore A = B is idempotent.</p><p>∴ Answer: A, B, C (Typically: A = B, A² = A, B² = B)</p>
Correct Answer: A,B,C