Matrix multiplication

In mathematics, matrix multiplication or the matrix product is a binary operation that produces a matrix from two matrices. The definition is motivated by linear equations and linear transformations on vectors, which have numerous applications 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 columns of $B$ and summed to produce an entry of $AB$ . When two linear transformations are represented by matrices, then the matrix product represents the composition of the two transformations.

The matrix product is not commutative in general, although it is associative and is distributive over matrix addition. The identity element of the matrix product is the identity matrix (analogous to multiplying numbers by 1), and a square matrix may have an inverse matrix (analogous to the multiplicative inverse of a number). Determinant multiplicativity applies to the matrix product. The matrix product is also important for matrix groups, and the theory of group representations and irreps.

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