Matrix Multiplication

Let’s learn Matrix Multiplication in Python 🧮✨—with clear explanations, step-by-step coding, and practice questions (with solutions)! 🚀

What is Matrix Multiplication? 🤔

• Matrix multiplication is a way to combine two matrices (2D lists) to produce a new matrix.
• You can multiply Matrix A (size m × n) with Matrix B (size n × p).
• The result is a new matrix of size m × p.

Rule: The number of columns in the first matrix must equal the number of rows in the second matrix!

Step-by-Step: Matrix Multiplication in Python 🐍

Let’s multiply two matrices step by step!

1️⃣ Example Matrices

A = [
    [1, 2, 3],
    [4, 5, 6]
]  # 2x3 matrix

B = [
    [7, 8],
    [9, 10],
    [11, 12]
]  # 3x2 matrix

• A is 2 rows × 3 columns
• B is 3 rows × 2 columns
• Result will be 2 rows × 2 columns

2️⃣ Algorithm (Logic)

• For each row in A,
  • For each column in B,
    • Multiply and sum the corresponding elements.

3️⃣ Code Implementation

def matrix_multiplication(X, Y):
    r1, c1 = len(X), len(X[^0])
    r2, c2 = len(Y), len(Y[^0])

    # Check if multiplication is possible
    if c1 != r2:
        print("Multiplication not possible!")
        return None

    # Initialize result matrix with zeros
    result = []
    for i in range(r1):
        result.append([^0] * c2)

    # Perform multiplication
    for i in range(r1):
        for j in range(c2):
            for k in range(c1):
                result[i][j] += X[i][k] * Y[k][j]
    return result

# Using the function
A = [
    [1, 2, 3],
    [4, 5, 6]
]
B = [
    [7, 8],
    [9, 10],
    [11, 12]
]
result = matrix_multiplication(A, B)
for row in result:
    print(row)
# Output:
# [58, 64]
# [139, 154]

📢 Explanation:

• result[i][j] is the dot product of the i-th row of A and the j-th column of B.
• The triple loop:
  • Outer loop: rows of A
  • Middle loop: columns of B
  • Inner loop: columns of A/rows of B (for multiplication and summing)

📝 Practice Questions with Solutions

Q1. Multiply the following matrices:

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

Solution:

result = matrix_multiplication(X, Y)
for row in result:
    print(row)
# Output:
# [6, 16]
# [7, 18]

Q2. What happens if you try to multiply a 2x3 matrix with a 2x2 matrix?

Answer: Multiplication is not possible because the number of columns in the first matrix (3) does not equal the number of rows in the second matrix (2).

🧩 Try It Yourself!

Q: Write a function to multiply two square matrices of size 3x3.

def multiply_3x3(A, B):
    result = [[^0]*3 for _ in range(3)]
    for i in range(3):
        for j in range(3):
            for k in range(3):
                result[i][j] += A[i][k] * B[k][j]
    return result

A = [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
]
B = [
    [9, 8, 7],
    [6, 5, 4],
    [3, 2, 1]
]
res = multiply_3x3(A, B)
for row in res:
    print(row)

💡 Key Points

• Matrix multiplication is only possible if the number of columns in the first matrix equals the number of rows in the second.
• Resulting matrix size: (rows of first) × (columns of second)
• Use nested loops to multiply and sum elements.

Now you can multiply matrices in Python! Try with your own numbers and see the magic! 🧙‍♂️✨

References:

• [Python-IITM-Foundational-Course.pdf]¹
• [Python-Cheatsheet-2024.pdf]²
• [Introduction_to_Python_Programming_-_WEB.pdf]³