Matrix Multiplication Explained - For Developers & ML Beginners

Matrix Multiplication Explained (The Black Box of Neural Networks)

Matrix multiplication is one of those operations that looks scary at first sight. But here’s the thing: without it, neural networks wouldn’t even exist. Today, we’re going to break it down step-by-step so you’ll never see it as a black box again.

Watch the video for a quick visual walkthrough, then read on for the details.

The Golden Rule of Matrix Multiplication

Before we do any math, there's one golden rule. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second. If they don't match, you can't multiply them. It's that simple.

Golden Rule about Matrix multiplication is that Cols of A should be equal to Rows of B

A (2x3) and (3x2) can be multiplied, but a (2x3) and (2x2) cannot.

The "Row-by-Column" Multiplication

The best way to think about the calculation is a "dance" between the rows of the first matrix and the columns of the second. Let's take two simple 2x2 matrices, A and B.

Matrix Multiplication

To find the top-left element of the result, we take the **first row of A** and the **first column of B**, multiply the corresponding elements, and add them up:

(1 × 5) + (2 × 7) = 19

We repeat this pattern for every element. For the top-right, it's the first row of A and the second column of B:

(1 × 6) + (2 × 8) = 22

After completing this for all positions, we get our final resulting matrix.

Matrix Multiplication

The Code: Using NumPy

In Python, we use NumPy's np.dot() method or the @ operator to do this.

import numpy as np

A = np.array([[1, 2],
              [3, 4]])

B = np.array([[5, 6],
              [7, 8]])

# Both of these methods do the same thing
C1 = np.dot(A, B)
C2 = A @ B

print(C1)
# Output:
# [[19 22]
#  [43 50]]

And remember the crucial point: with matrices, order matters. A @ B is not the same as B @ A.

Conclusion

Matrix multiplication is the fundamental operation that transforms data in a neural network. It's how raw inputs are combined with learned "weights" to produce meaningful patterns.

Your Turn...

Do you find it easier to think of matrix multiplication as rows × columns, or as a way to combine patterns? Share your thoughts in the comments!

This post is part of the "Decoding the Math of ML" series. For the previous part, check out: Matrices in Machine Learning.

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