Matrix Multiplication - 2

Let’s dive deeper into matrix multiplication 2 in Python! 🧮✨ We’ll cover both the step-by-step logic and a modular, function-based approach—with practice and solutions! 🚀

🟢 What is Matrix Multiplication?

• Multiply Matrix A (size m × n) with Matrix B (size n × p).
• The result is a new matrix of size m × p.
• Each element in the result is the dot product of a row from A and a column from B.

🟡 Modular Matrix Multiplication Using Functions

A clean way to multiply matrices is to use helper functions for:

• Getting a row from a matrix
• Getting a column from a matrix
• Calculating the dot product of two lists

Let’s see this in action! 👇

Step 1: Helper Functions

# Get the ith row
def row(M, i):
    return M[i]

# Get the jth column
def column(M, j):
    return [M[k][j] for k in range(len(M))]

# Dot product of two lists
def dot(u, v):
    return sum(u[k] * v[k] for k in range(len(u)))

Step 2: Matrix Multiplication Function

def matmul(A, B):
    n = len(A)  # assuming square matrices of size n x n
    C = []
    for i in range(n):
        row_result = []
        for j in range(n):
            rowA = row(A, i)
            colB = column(B, j)
            row_result.append(dot(rowA, colB))
        C.append(row_result)
    return C

Step 3: Example Usage

A = [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
]
B = [
    [1, 2, 1],
    [3, 1, 7],
    [6, 2, 3]
]
result = matmul(A, B)
for r in result:
    print(r)
# Output:
# [25, 12, 28]
# [55, 27, 64]
# [85, 42, 100]

How does it work?

• For each cell in the result, we take the corresponding row from A and column from B, compute their dot product, and store it.

🟠 Practice Question

Q: Multiply the following 2x2 matrices using the modular function approach:

X = [
    [2, 4],
    [3, 4]
]
Y = [
    [1, 2],
    [1, 3]
]

def row(M, i):
    return M[i]

def column(M, j):
    return [M[k][j] for k in range(len(M))]

def dot(u, v):
    return sum(u[k] * v[k] for k in range(len(u)))

def matmul(A, B):
    n = len(A)
    C = []
    for i in range(n):
        row_result = []
        for j in range(n):
            rowA = row(A, i)
            colB = column(B, j)
            row_result.append(dot(rowA, colB))
        C.append(row_result)
    return C

X = [
    [2, 4],
    [3, 4]
]
Y = [
    [1, 2],
    [1, 3]
]
result = matmul(X, Y)
for r in result:
    print(r)
# Output:
# [6, 16]
# [7, 18]

🔵 Key Points

• Matrix multiplication can be broken into smaller functions: row, column, dot product.
• This modular approach makes your code clear and easy to debug! 🛠️
• Always check that the number of columns in the first matrix equals the number of rows in the second.

Try it with your own matrices and see the magic! ✨ If you want to multiply non-square matrices, just adjust the loops to use len(A) for rows and len(B) for columns¹.

References:

• Python-IITM-Foundational-Course.pdf (modular matrix multiplication using functions)¹
• Python-Cheatsheet-2024.pdf (step-by-step multiplication logic)²