Determináns

Determináns (determinant) is a scalar attribute of a square matrix. For an n×n matrix A over a field, det(A) is denoted det(A) or |A| and is zero exactly when A is singular. Geometrically, det(A) gives the factor by which A scales n-dimensional volumes under the associated linear transformation; its absolute value is the volume scale, while the sign indicates orientation preservation or reversal.

Computation: For a 2×2 matrix A = [[a, b], [c, d]], det(A) = ad − bc. For larger matrices, determinants can be computed by expansion along a row or column (cofactor expansion) or, more efficiently, by reducing to upper triangular form using row operations and taking the product of the diagonal entries, counting row swaps (which flip the sign) and any row-scale factors.

Properties and applications: det(AB) = det(A) det(B). det(A^T) = det(A). det(kA) = k^n det(A) for an n×n matrix. If A is invertible, det(A^{-1}) = 1/det(A). det(A) = 0 iff A is singular. Cramer's rule uses determinants to solve linear systems; the characteristic polynomial det(A − λI) yields eigenvalues. Geometrically, det measures how volumes change under the linear map and the sign indicates orientation; in graphics and physics, determinants appear in coordinate changes and volume computations.