determinaten

The determinant is a scalar attribute of a square matrix that encodes how a linear transformation stretches or shrinks volumes and whether it preserves or reverses orientation. For an n by n matrix, the determinant gives the factor by which the transformation scales the n-dimensional volume of a geometric object, such as a parallelepiped formed by the matrix’s columns (or rows). If the determinant is zero, the transformation collapses volume in at least one dimension and the matrix is singular.

Notation and simple cases: the determinant is denoted det(A) or |A|. For a 2×2 matrix [a b; c d], the determinant is ad − bc. For larger matrices, the determinant can be computed via expansion by minors (cofactors), or by row reduction, or by a permutation formula. In general, the determinant is a polynomial in the matrix entries of degree n.

Key properties: the determinant is multilinear in the rows (or columns) and alternating, so swapping two rows negates det(A). Multiplying a row by a scalar c multiplies det(A) by c, and det(AB) = det(A) det(B). The determinant of the transpose equals the determinant, det(A^T) = det(A). The determinant of the identity is 1, and det(A) = 0 if and only if A is singular (non-invertible). If A is invertible, det(A^{-1}) = 1/det(A).

Computation and interpretation: common methods include expansion by cofactors and row-reduction techniques that track pivot multipliers and row swaps. Applications include solving systems of linear equations (Cramer's rule applies when det(A) ≠ 0), finding eigenvalues through the characteristic polynomial det(A − λI) = 0, and measuring volume distortion and orientation change induced by a linear map.

History: determinants were developed in the 18th century by mathematicians such as Leibniz and Cauchy; the term reflects the function’s role in determining invertibility and volume changes.