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If A and B are two matrices of order 3 × 3 satisfying A B = A and B A = B , then A + B 5 is equal to

Options

  1. A5 A + B
  2. B5 I
  3. C16 A + B
  4. D32 I

Correct answer

C. 16 A + B

Step-by-step solution

A B = A , B A = B (Given) ∴ A 2 = A ⋅ A = A B A = A B A = A B = A ⇒ A 5 = A 4 = A 3 = A Similarly, B 2 = B ⋅ B = B A ⋅ B = B A B = B A = B ⇒ B 5 = B 4 = B 3 = B So, A + B 2 = A 2 + B 2 + A B + B A = 2 A + B A + B 3 = A + B 2 A + B = 2 A + B 2 = 4 A + B Similarly, A + B 4 = 8 A + B A + B 5 = 16 A + B

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