Matrix multiplication

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In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. The resulting matrix agrees with the result of composition of the linear transformations represented by the two original matrices.

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Matrix product

The matrix product is the most commonly used type of product of matrices. Matrices offer a concise way of representing linear transformations between vector spaces, and matrix multiplication corresponds to the composition of linear transformations. The matrix product of two matrices can be defined when their entries belong to the same ring, and hence can be added and multiplied, and, additionally, the number of the columns of the first matrix matches the number of the rows of the second matrix. The product of an m×p matrix A with an p×n matrix B is an m×n matrix denoted AB whose entries are

where 1 ≤ im is the row index and 1 ≤ jn is the column index. This definition can be restated by postulating that the matrix product is left and right distributive and the matrix units are multiplied according to the following rule:

where the first factor is the m×n matrix with 1 at the intersection of the ith row and the kth column and zeros elsewhere and the second factor is the p×n matrix with 1 at the intersection of the lth row and the jth column and zeros elsewhere.

Properties

  • In general, matrix multiplication is not commutative. More precisely, AB and BA need not be simultaneously defined; if they are, they may have different dimensions; and even if A and B are square matrices of the same order n, so that AB and BA are also square matrices of order n, if n is greater or equal than 2, AB need not be equal to BA. For example,

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