Why is matrix multiplication defined by this particular rule?

The rule of matrix multiplication is based on the concept of linear transformations and composition of functions.

If matrices are viewed as operators acting on vectors, then multiplying matrices corresponds to applying these operators sequentially.

An element of the result matrix at position (i, j) is obtained as the sum of products of elements from the i-th row of the first matrix and the j-th column of the second matrix. This reflects how each component of the input vector is transformed and summed to produce the output vector.

This rule ensures consistency with algebraic properties and allows matrices to be used for describing systems of linear equations, transformations in space, and other tasks.

Example in Go:

package main
import "fmt"

func multiply(a, b [][]int) [][]int {
    n, m, p := len(a), len(b[0]), len(b)
    result := make([][]int, n)
    for i := range result {
        result[i] = make([]int, m)
        for j := 0; j < m; j++ {
            sum := 0
            for k := 0; k < p; k++ {
                sum += a[i][k] * b[k][j]
            }
            result[i][j] = sum
        }
    }
    return result
}

func main() {
    a := [][]int{{1, 2}, {3, 4}}
    b := [][]int{{5, 6}, {7, 8}}
    c := multiply(a, b)
    fmt.Println(c) // [[19 22] [43 50]]
}