Determinanti

Determinanti are scalar values associated with square matrices that encapsulate several important algebraic properties of the matrix. Formally, the determinant of an n × n matrix A, denoted det(A) or |A|, is a function mapping A to a real or complex number, defined recursively by expansion along a row or column, or equivalently by the Leibniz formula summing signed products of entries over all permutations of {1,…,n}. The determinant is multilinear in the rows (or columns) and alternating: swapping two rows changes its sign, and a matrix with two equal rows has determinant zero.

Key properties include multiplicativity (det(AB)=det(A)·det(B)), invariance under transposition (det(Aᵀ)=det(A)), and the relationship with invertibility: a matrix is invertible if and only if its determinant is non‑zero, and the inverse can be expressed using the adjugate matrix divided by the determinant. Determinants also give the scaling factor of the linear transformation represented by the matrix, measuring how volumes change under the transformation; a positive determinant preserves orientation, while a negative one reverses it.

Computational methods range from cofactor expansion for small matrices to row‑reduction techniques such as Gaussian elimination, which reduce the matrix to triangular form, making the determinant the product of diagonal entries up to sign adjustments for row swaps. Specialized algorithms like LU decomposition improve efficiency for large systems.

Historically, determinants appeared in the works of Seki Kōwa and Leibniz in the 17th century, with systematic development by Cramer, Vandermonde, and later by Cayley and Sylvester. Today they are fundamental in linear algebra, differential equations, geometry, and physics, where they appear in eigenvalue problems, change‑of‑variables formulas, and the formulation of conservation laws.