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Let P = 1 0 0 4 1 0 16 4 1 and I be the identity matrix of order 3 . If Q = q i j is a matrix such that P 50 - Q = I , then the value of q 31 + q 32 q 21 is equal to

Options

  1. A52
  2. B103
  3. C201
  4. D205

Correct answer

B. 103

Step-by-step solution

P = 1 0 0 4 1 0 16 4 1     ⇒ P 2 = 1 0 0 8 1 0 16 + 32 8 1 So, P 3 = 1 0 0 12 1 0 16 + 32 + 48 12 1 Using symmetry, we can write P 50 = 1 0 0 200 1 0 16.50.51 2 200 1 As, P 50 − Q = I ⇒ Q = P 50 − I = [ 0 0 0 200 0 0 16.50.51 2 200 0 ]   , q 31 = 16.50.51 2 , q 32 = 200 and q 21 = 200 ∴ q 31 + q 32 q 21 = 16.50.51 2.200 + 1 = 102 + 1 = 103

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