Prove that A is idempotent if and only if AT is idempotent. Getting Started: The

Question

Prove that A is idempotent if and only if AT is idempotent. Getting Started: The phrase "if and only if" means that you have to prove two statements. If A is idempotent, then AT is idempotent. If AT is idempotent, then A is idempotent. Begin your proof of the first statement by assuming that A is idempotent. This means that A2 = Take the transpose of both sides of the equation from Step 1. Use the properties of the transpose to simplify your result from Step 2. This shows that AT is idempotent. Begin your proof of the second statement by assuming that AT is idempotent. This means that AT Take the transpose of both sides of the equation from Step 4. Use the properties of the transpose to simplify your result from Step 5. This shows that A is idempotent.

Explanation / Answer

STEP1: A^2 = A

STEP2: (A^2)^T = A^T

STEP3: A^T A^T = A^T

STEP4: A^T = A^T A^T

STEP5: (A^T)^T = (A^T A^T)^T

STEP6: A = (A^T)^T (A^T)^T

          A = A A

        A = A^2

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