Matrixprodukt

Matrixprodukt, or matrix product, is a binary operation that takes two matrices and produces a third. It is defined when the number of columns of the first matrix equals the number of rows of the second; if A is m×n and B is n×p, the product AB is an m×p matrix.

Entries are given by (AB)_{ij} = sum_{k=1}^n A_{ik} B_{kj}. In linear algebra, matrices represent linear maps, and the matrix product AB corresponds to the composition of the map B followed by A.

Key properties include associativity (A(BC) = (AB)C) and distributivity over addition (A(B+C) = AB + AC, (A+B)C = AC + BC). The identity matrix I_m acts as a left and right identity when the dimensions match: I_m A = A and A I_n = A. For square matrices, det(AB) = det(A) det(B); if A and B are invertible, (AB)^{-1} = B^{-1} A^{-1}.

Computation is performed by row-by-column dot products: the i-th row of A is multiplied by the j-th column of B. The operation is not commutative in general, so the order AB matters.

Matrix multiplication underlies many applications, including composition of linear transformations, coordinate changes, computer graphics transformations, and the solution of linear systems via Ax = b. As a simple illustration, with A = [[1,2],[3,4]] and B = [[5,6],[7,8]], AB = [[19,22],[43,50]].