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.
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- Definition of multilinear functionA function of several vector arguments is multilinear if it is linear in each argument while the others are held...
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- The determinant function is unique.An alternating multilinear function of $n$ columns with value $1$ on the standard basis is unique. Expand each column...
- A determinant is a multilinear functionThe determinant is linear in one row or column with every other row or column fixed . For example, $\det(u+v,c...
- Adding a multiple of one row to another row does not change the determinant.Replacing row $i$ by $R i+cR j$, with $i\ne j$, preserves the determinant. By multilinearity, the new determinant is...
- Switching two rows multiplies the determinant by -1.Interchanging two rows or two columns changes the determinant& 39;s sign. This follows from the alternating property....
- Multiplying a row by a scalar multiplies the determinant by that scalar.Scaling a single row by $c$ scales the determinant by $c$, by linearity in that row. Scaling all rows gives...
- A matrix with two equal rows/columns has determinant 0A matrix with two equal rows or columns has determinant zero, by the alternating property. More generally, linearly...
- A matrix with a 0 row/column has determinant 0A square matrix with a zero row or column has determinant zero: by multilinearity, replacing one argument by the zero...
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