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Manipal MET2013MathematicsMatrices

For two unimodular complex numbers z₁ and z₂, [ array ll z ₁ & z₂ z ₂ & z₁ array ]⁻¹ [ array cc z₁ & z₂ - z ₂ & z ₁ array ]⁻¹ is equal to

Options

  1. A[ array ll z₁ & z₂ z ₁ & z ₂ array ]
  2. B[ array ll 1 & 0 0 & 1 array ]
  3. C[ array cc 1 / 2 & 0 0 & 1 / 2 array ]
  4. DNone of these

Correct answer

C. [ array cc 1 / 2 & 0 0 & 1 / 2 array ]

Step-by-step solution

aligned & [ array cc z ₁ & -z₂ z ₂ & z₁ array ]⁻¹ [ array cc z₁ & z₂ - z ₂ & z ₁ array ]⁻¹ &= [ array cc z₁ & z₂ - z ₂ & z ₁ array ] [ array cc z ₁ & -z₂ z ₂ & z₁ array ] ⁻¹ &= [ array cc z₁ z ₁+z₂ z ₂ & 0 0 & z₂ z ₂+z₁ z ₁ array ]⁻¹ &= [ array cc |z₁ |^2+ |z₂ |^2 & 0 0 & |z₁ |^2+ |z₂ |^2 array ]⁻¹ &= [ array cc 2 & 0 0 & 2 array ]⁻¹= [ array cc 1 / 2 & 0 0 & 1 / 2 array ] aligned

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