matrice

A matrice, in mathematics, is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is typically denoted A and has m rows and n columns (an m×n matrice). Entries are a_ij, where i ranges over rows and j over columns. Matrices over a field, usually the real or complex numbers, encode linear transformations and systems of linear equations.

Operations include addition and scalar multiplication (elementwise), and matrix multiplication, defined by (AB)_{ij} = sum_k a_ik b_kj. The transpose A^T switches rows and columns. For square matrice, the determinant det(A) measures volume scaling; det(A) ≠ 0 implies invertibility, with inverse A^{-1} defined by AA^{-1}=I. The rank is the dimension of the row or column space.

Special matrice include the identity I_n, the zero matrix, diagonal, triangular, symmetric (A^T = A), Hermitian in complex cases (A^H = A), and orthogonal (A^T A = I). Eigenvalues and eigenvectors solve Av = λv, revealing invariant directions under the transformation. Factorizations such as LU, QR, and singular value decomposition (A = UΣV^T) underpin numerical methods for solving systems and computing inverses.

Applications include solving linear systems, computer graphics and geometric transformations, statistics (covariance matrices), physics, and machine learning. The term matrix was introduced in the 19th century by James Sylvester, from Latin matrix meaning "womb" or "source," reflecting its role as a building block for linear mappings.