matrixa

Matrixa is a term used in mathematics to denote a structured array of numbers, functions, or other objects arranged in rows and columns, similar in purpose to a matrix. In formal style, a matrixa is an m-by-n array A = [a_ij], where i ranges from 1 to m and j from 1 to n, with entries a_ij drawn from a field or ring such as the real numbers or complex numbers.

Operations on matrixa mirror those of standard matrices. Addition and scalar multiplication are performed elementwise. Product of two matrixa A and B is defined when the inner dimensions agree: if A is m x p and B is p x n, the product AB is an m x n matrixa with entries (AB)_ij = sum_{k=1}^p a_ik b_kj.

For square matrixa (m = n), the determinant det(A) is defined, and a matrixa is invertible when det(A) ≠ 0. The transpose A^T interchanges rows and columns. Many special types exist, including diagonal, symmetric, and orthogonal matrixa, each with characteristic properties useful in theory and computation.

Matrixa are used to model linear transformations, solve systems of linear equations, and perform data analysis in science and engineering. They provide a compact language for describing rotations, projections, and other linear operations in vector spaces.

Example: A = [ [1, 2, 3], [4, 5, 6] ] is 2 x 3 and B = [ [7, 8], [9, 10], [11, 12] ] is 3 x 2; AB = [ [58, 64], [139, 154] ].

See also: Matrix, Linear algebra, Determinant, Inverse, Matrix multiplication.