Chapter 1: Calculate
Matrix Multiplication by Hand ✍️ Workbook
I didn’t make this workbook because I immediately understood matrix multiplication or fell in love with it. Quite the opposite—I struggled with it for a long time.
Matrix multiplication felt abstract, mechanical, and easy to get wrong. I could follow procedures, but I didn’t really see what was happening or why it worked. Over time, as I tried to teach this topic to my students, I realized that many of them were stuck in the same place I had been.
This workbook grew out of that struggle. The approach here is the method I gradually discovered—by slowing things down, working through matrix multiplication by hand ✍️, and forcing myself to reason about shapes, operations, and structure one step at a time. I refined this method over several years of teaching, revising it whenever something didn’t click for my students.
Three moments changed everything for me.
1. The first was realizing that matrix multiplication is just lots of dot products. Each entry in the result comes from a single dot product—nothing more.
2. The second was understanding rows × columns: every output value comes from pairing one row from the left matrix with one column from the right matrix. This made shapes and dimensions finally make sense.
3. The third was noticing that if I shift the columns to the top, I can visually see where every row and column intersect. That simple rearrangement made the shape of the output matrix immediately clear, before doing any calculation.
The chapters are organized in a deliberate, linear order. I recommend working through them sequentially. Most pages use fill-in-the-blank exercises to encourage active thinking. Try each problem on your own first, then check the solutions to reflect on your reasoning and correct misunderstandings.
My hope is that this workbook helps you build the intuition I wish I had earlier—an intuition for matrix multiplication that feels concrete, reliable, and usable in real AI systems.
The rest of the chapter (blank + solution) is available to AI by Hand Academy members. You can become a member via a paid Substack subscription.