Determinantenregels

Determinantenregels, or determinant rules, are a set of algebraic properties of determinants used in linear algebra to simplify computation, reason about invertibility, and interpret volume changes under linear maps.

The determinant of an n-by-n matrix A is a scalar that encodes whether A is invertible (det(A) ≠ 0) and the oriented volume of the parallelepiped spanned by its rows (or columns). It is a fundamental invariant under many matrix operations.

Key properties include multilinearity in the rows, meaning the determinant is linear in each row separately when the other rows are kept fixed, and the alternating property, which implies that swapping two rows changes the sign and that equal rows yield det(A) = 0. These ideas underpin many determinant calculations and theoretical results.

Elementary row operations affect the determinant in specific ways: adding a multiple of one row to another does not change det; swapping two rows multiplies det by -1; multiplying a row by a nonzero scalar c multiplies det by c. These rules explain how row reduction preserves or changes the determinant during computation.

Other rules include det(AB) = det(A) det(B), and det(A^T) = det(A). If A is invertible, det(A^{-1}) = 1 / det(A). Cofactor expansion allows computing det(A) by expanding along a row or column. For special matrices, the determinant is particularly easy to compute: the determinant of a diagonal or triangular matrix equals the product of its diagonal entries; for block diagonal matrices, the determinant is the product of the determinants of the blocks.

Interpretation: a zero determinant indicates a degenerate linear transformation (volume zero), while the magnitude of the determinant measures the volume scaling induced by the transformation.