m×nmatris

A m×n matris, or m-by-n matrix, is a rectangular array of elements arranged in m rows and n columns. Each element is denoted aij, where i = 1,...,m is the row index and j = 1,...,n is the column index. The size is called m-by-n. Elements can be real numbers, complex numbers, or elements of any field. Common special cases include row vectors (1×n) and column vectors (m×1). For example, A = [ [1, 2, 3], [4, 5, 6] ] is a 2×3 matrix.

Operations on matrices include addition and subtraction, which are performed elementwise on matrices of the same size. Scalar multiplication multiplies every entry by a scalar. Matrix multiplication is defined when the inner dimensions match: if A is m×n and B is n×p, then AB is defined and has size m×p, with entries computed as (AB)ij = sum over k of aik bkj. In general, AB does not equal BA, and matrix multiplication is not commutative.

The transpose of a matrix A, denoted AT, is obtained by swapping rows and columns, yielding an n×m matrix with (AT)ij = aji. The rank of a matrix is the maximum number of linearly independent rows (or columns) and satisfies 0 ≤ rank(A) ≤ min(m, n). Only square matrices (m = n) can be invertible; a matrix A is invertible if there exists B with AB = BA = I, where I is the identity matrix. Determinants are defined for square matrices and provide a criterion for invertibility and volume scaling under linear transformations.

Applications of m×n matrices include solving systems of linear equations Ax = b, representing linear transformations from one vector space to another, and numerous uses in science, engineering, and data analysis.