m×nmatrise

A m×n matrix, often written as A = (a_ij), is a rectangular array with m rows and n columns. Each a_ij is an element from a chosen set, typically a field such as the real numbers or complex numbers. Matrices are used to represent linear mappings, systems of equations, and data in a compact form.

Matrices can be added or scaled when they have the same dimensions. For two m×n matrices A and B, their sum is the matrix (a_ij + b_ij). Scalar multiplication multiplies every entry by a chosen number. Matrix multiplication is defined when A is m×p and B is p×n, yielding a product C = AB that is m×n. Each entry of C is computed as C_ij = sum_{k=1}^p A_ik B_kj.

The transpose of a matrix A, denoted A^T, is the n×m matrix with entries (A^T)_ji = a_ij. The rank of A is the dimension of its row space or column space and satisfies rank ≤ min(m, n). Rank can be found through row-reduction to echelon form and is central to solving linear systems.

Determinants are defined only for square matrices. If A is n×n and det(A) ≠ 0, then A is invertible and has a unique inverse A^{-1}. Non-square matrices do not have a two-sided inverse, though left or right inverses may exist if A has full row or full column rank. The identity matrix I_n is the n×n matrix with ones on the diagonal and zeros elsewhere.

Applications of m×n matrices include solving linear systems Ax = b, representing linear transformations from R^n to R^m, data representation in statistics and computer graphics, and, for square matrices, analysis via eigenvalues and the determinant.

Example: A = [[1, 2, 3], [4, 5, 6]] is a 2×3 matrix.