Python Matrix Multiplication — 3 Methods

What Is Matrix Multiplication?

A matrix is a 2D data structure where numbers are arranged into rows and columns. Python has no built-in matrix type — nested lists represent matrices.

The matrix product (dot product) of matrices A and B computes: result[i][j] = sum(A[i][k] * B[k][j] for all k)

Example: A = [[1,2],[3,4]], B = [[1,3],[2,5]]
Result = [[5,13],[11,29]]

Requirement: columns of A must equal rows of B. An (m×n) matrix times an (n×p) matrix gives an (m×p) result.

Approach 1: Nested Loops

The most explicit approach — three nested loops iterate over every element combination:

def Multiply(A, B):
    result = [[0,0,0], [0,0,0], [0,0,0]]
    for i in range(len(A)):
        for j in range(len(B[0])):
            for k in range(len(B)):
                result[i][j] += A[i][k] * B[k][j]

    for p in result:
        print(p)
    return 0

A = [[1, 2, 3], [6, 7, 4], [8, 10, 11]]
B = [[1, 5, 3], [2, 6, 5], [7, 4, 9]]

print("Result:")
Multiply(A, B)

# Output:
# Result:
# [26, 29, 40]
# [48, 88, 89]
# [105, 144, 173]

Approach 2: List Comprehension

More concise — compresses the three loops into a single expression using zip and sum:

def Multiply(X, Y):
    result = [[sum(a*b for a,b in zip(X_row, Y_col))
               for Y_col in zip(*Y)]
              for X_row in X]
    for k in result:
        print(k)
    return 0

A = [[6, 7, 2], [3, 5, 4], [1, 2, 3]]
B = [[1, 5], [2, 5], [6, 3]]

print("Result:")
Multiply(A, B)

# Output:
# Result:
# [32, 71]
# [37, 52]
# [23, 24]

zip(*Y) transposes B — turning columns into rows so zip(X_row, Y_col) pairs matching elements. sum(a*b ...) computes the dot product for each output cell.

Approach 3: NumPy

For any production or numerical computing use, NumPy's dot() is the right tool — implemented in optimized C/Fortran (BLAS), orders of magnitude faster than pure Python loops. It's worth internalizing: every neural network forward pass — from CNN convolutions to transformer attention — reduces to exactly this operation, executed billions of times:

import numpy as np

A = [[12, 7, 3], [4, 5, 6], [7, 8, 9]]
B = [[5, 8, 1, 2], [6, 7, 3, 0], [4, 5, 9, 1]]

result = np.dot(A, B)   # or: A @ B (Python 3.5+)

for p in result:
    print(p)

# Output:
# [114 160  60  27]
# [ 74  97  73  14]
# [119 157 112  23]