Matrix multiply

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring. The matrix product is designed for representing the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, physics, and engineering. In more detail, if $A$ is an $n \times m$ matrix and $B$ is an $m \times p$ matrix, their matrix product $AB$ is an $n \times p$ matrix, in which the $m$ entries across a row of $A$ are multiplied with the $m$ entries down a column of $B$ and summed to produce an entry of $AB$ . When two linear maps are represented by matrices, then the matrix product represents the composition of the two maps.

The definition of matrix product requires that the entries belong to a ring, which may be noncommutative, but is a field in most applications. Even in this latter case, matrix product is not commutative in general, although it is associative and is distributive over matrix addition. The identity matrices (which are the square matrices whose all entries are zero, except those of the main diagonal that are all equal to 1) are identity elements of the matrix product. It follows that the $n \times n$ matrices over a ring form a ring, which is noncommutative except if $n = 1$ and the ground ring is commutative.

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