matricës

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is typically denoted by a capital letter, such as A, and its size is written as m by n (m×n), where m is the number of rows and n the number of columns. The entry in the i-th row and j-th column is denoted aij. Matrices are used to represent linear transformations and to organize data.

There are many special kinds of matrices. A square matrix has the same number of rows and columns (m = n). Diagonal matrices have nonzero entries only on the main diagonal. The identity matrix I has ones on the main diagonal and zeros elsewhere. Symmetric matrices satisfy A^T = A, while skew-symmetric matrices satisfy A^T = −A. Orthogonal matrices meet A^T A = A A^T = I, preserving lengths and angles. The determinant is defined for square matrices and indicates invertibility and volume scaling under the associated linear transformation. A matrix is invertible if det(A) ≠ 0, in which case A^{-1} exists and A A^{-1} = A^{-1} A = I. The transpose A^T swaps rows with columns, and the rank measures the dimension of the row or column space.

Matrix multiplication combines two matrices when the inner dimensions match: A (m×n) times B (n×p) yields a matrix C (m×p). This operation corresponds to composing linear maps. Eigenvalues and eigenvectors are defined for square matrices and reveal intrinsic directions of stretching.

Applications span solving linear systems (Gaussian elimination), representing linear transformations in geometry, computer graphics, statistics (covariance matrices), and machine learning (weight and feature matrices). The concept originated in the 19th century, with contributions from James Joseph Sylvester and Arthur Cayley, and remains foundational across disciplines.