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If B is an idempotent matrix and A is defined by A = I − B, which of the following is true?

1. A² = A
2. A² = I
3. AB = I
4. BA = I

Correct answer: A² = A

Solution

An idempotent matrix B satisfies the property B² = B. Therefore, when we compute A² = (I - B)(I - B), we can expand this to I - 2B + B². Since B² = B, this simplifies to I - 2B + B = I - B, which is equal to A. Thus, A² = A.

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