Determinants axiomatically
The determinant can be defined axiomatically as the unique function $\det: M_n(\mathbb{F}) \to \mathbb{F}$ satisfying:
- Multilinearity: linear in each row (column)
- Alternating: swapping two rows changes sign
- Normalization: $\det(I) = 1$
These three properties uniquely characterize the determinant and can be used to derive all other properties.
Parents
Children
- Definition of multilinear function
- The determinant function exists.
- The determinant function is unique.
- A determinant is a multilinear function
- Adding a multiple of one row to another row does not change the determinant.
- Switching two rows multiplies the determinant by -1.
- Multiplying a row by a scalar multiplies the determinant by that scalar.
- A matrix with two equal rows/columns has determinant 0
- A matrix with a 0 row/column has determinant 0