Matrix multiplication is a fundamental operation in linear algebra, involving the combination of two matrices to produce a new matrix. Given two matrices, A (of size m × n) and B (of size n × p), the resulting matrix C (of size m × p) is computed by taking the dot product of the rows of matrix A with the columns of matrix B.
The elements of the resulting matrix C are calculated using the following formula:
Cij = Σ (Aik * Bkj) for k = 1 to n
This means that each element Cij is the sum of the products of corresponding elements from the i-th row of matrix A and the j-th column of matrix B.
Matrix multiplication is not commutative; that is, A × B is not necessarily equal to B × A. However, it is associative and distributive over addition.
Matrix multiplication has extensive applications in various fields, including computer graphics, machine learning, and systems of linear equations. In the context of AI, it is particularly important for operations involving neural networks, where layers of neurons can be represented as matrices, allowing for efficient computation of outputs through transformations.