Matrix multiplication is an operation that combines two matrices to produce a resulting matrix. It is a fundamental concept in linear algebra and is commonly used in various fields such as computer science, physics, and economics.

  To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, also known as the product matrix, will have the same number of rows as the first matrix and the same number of columns as the second matrix.

  The multiplication process involves multiplying corresponding elements in each row of the first matrix with the corresponding elements in each column of the second matrix. These products are then summed up to find the respective elements of the resulting matrix.

  For example, if we have two matrices A and B with dimensions m x n and n x p, respectively, the resulting matrix C will have dimensions m x p. Each element cij in the resulting matrix C is calculated as the sum of the products of corresponding elements in the ith row of matrix A and the jth column of matrix B:

  cij = a1i * b1j + a2i * b2j + ... + ani * bnj

  Matrix multiplication is not commutative, meaning that the order of multiplication matters. In general, AB is not equal to BA if A and B are matrices of appropriate dimensions.

  Matrix multiplication plays a crucial role in solving systems of linear equations, transformations, finding eigenvalues and eigenvectors, and many other applications in various disciplines.