This article will use the following notational conventions: matrices are represented by capital letters in bold, e.g. A; See more

If A is an m × n matrix and B is an n × p matrix,
That is, the entry of the product is obtained by multiplying term-by-term the entries of the ith row of A and the jth column of B, and summing th… See more

Historically, matrix multiplication has been introduced for facilitating and clarifying computations in linear algebra. This strong relationship between matrix multiplication and linear algebra remains fundamental in all mathem… See more

Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second fact… See more

Let us denote the set of n×n square matrices with entries in a ring R, which, in practice, is often a field.
In , the product is defined for every pair of matrices. This makes a ring, which has the identity matrix I … See more

The definition of matrix product requires that the entries belong to a semiring, and does not require multiplication of elements of the semiring to be commutative. In many applications, the matrix elements belong to a field, alth… See more

Other types of products of matrices include:
• Block matrix operations
• Cracovian product, defined as A ∧ B = B A
• Frobenius inner product, the dot product of matrices considered as vectors, or, equivalently the sum of the … See more