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If A = 1 5 2 5 - 2 5 1 5 , B = 1 0 i 1 , i = - 1 , and Q = A T B A , then the inverse of the matrix A Q 2021 A T is equal to:

Options

  1. A1 − 2021 0 1
  2. B1 0 - 2021 i 1
  3. C1 5 − 2021 2021 1 5
  4. D1 0 2021 i 1

Correct answer

B. 1 0 - 2021 i 1

Step-by-step solution

AA ⊤ = 1 5 2 5 - 2 5 1 5 1 5 - 2 5 2 5 1 5 = 1 0 0 1 = I Q = A ⊤ B A ⇒ Q 2 = A ⊤ B A A ⊤ B A = A ⊤ B A A ⊤ B A = A ⊤ B I B A = A ⊤ B 2 A ⇒ Q 3 = A ⊤ B 3 A ⇒ Q 2021 = A T B 2021   A Now let P = AQ 2021   A T P = A A ⊤ B 2021   A A ⊤ given A A ⊤ = I P = B 2021 B 2 = 1 0 i 1 1 0 i 1 = 1 0 2 i 1 B 3 = 1 0 2 1 1 0 i 1 = 1 0 3 i 1 B 2021 = 1 0 2021 i 1 inverse of P = P - 1 = B 2021 - 1 = 1 0 - 2021 i 1

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