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Let A and B are two non-singular matrices such that A B = B A 2 ,   B 4 = I and A k = I , then k can be equal to

Options

  1. A5
  2. B10
  3. C15
  4. D16

Correct answer

C. 15

Step-by-step solution

A B 4 = A B ( B ) ( B ) ( B ) = B A 2 ( B ) ( B ) ( B ) = B A ( A B ) B B = B A ( B A 2 ) B B = B ( A B ) ( A 2 B B ) = B B A 4 B B = B 2 B A 8 B = B 4 A 16 ⇒ A = A 16 ⇒ A 15 = I

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