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[QUE/VS-08001]

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On the vector space of \(n\times n\) complex matrices, which of the following define scalar products and which one does not? Give a short explanation for your answers.

  1. \( (x,y)= \text{Tr}(x^\dagger,y)\)
  2. \((x,y)= \text{Tr}(x) + \text{Tr}(y)\)
  3. \((x,y)= \det(x^\dagger y)\)

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