Determinanten

Determinant is a scalar associated with a square matrix A, denoted det(A) or |A|. For a 1x1 matrix [a], det(A) = a. For larger matrices, the determinant provides a criterion for invertibility: det(A) = 0 iff A is singular. Geometrically, it represents the factor by which the linear transformation associated with A expands or contracts volumes, and its sign reflects orientation.

Computation: For a 2x2 matrix [[a,b],[c,d]], det = ad - bc. In general, determinants can be computed by cofactor expansion, row reduction, or LU decomposition. Elementary row operations affect det: adding a multiple of one row to another leaves det unchanged; swapping rows multiplies det by -1; multiplying a row by a scalar c multiplies det by c. In practice, the determinant of a product equals the product of determinants, and the determinant of a triangular matrix is the product of its diagonal entries.

Key properties: det(AB) = det(A) det(B); det(A^T) = det(A); det(kA) = k^n det(A) for an n-by-n matrix; det(I) = 1; det(0) = 0. A matrix is invertible exactly when det(A) ≠ 0. The determinant is the product of the eigenvalues (counting multiplicities).

Applications: Cramer's rule expresses solutions of linear systems in terms of determinants if det(A) ≠ 0. In calculus, the Jacobian determinant appears in change of variables for integrals. In geometry, the absolute value of det(A) gives the volume scaling factor of the linear transformation, and the sign indicates orientation.