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Determinants axiomatically

Created over 8 years ago, updated 10 days ago

The determinant can be defined axiomatically as the unique function $\det: M_n(\mathbb{F}) \to \mathbb{F}$ satisfying:

  1. Multilinearity: linear in each row (column)
  2. Alternating: swapping two rows changes sign
  3. Normalization: $\det(I) = 1$

These three properties uniquely characterize the determinant and can be used to derive all other properties.