Composition

Matrix multiplication, and why order matters

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $AB$ means "do $B$, then $A$". Entries are rows of $A$ dotted with columns of $B$.
• In general $AB \ne BA$: order matters.
• $\det(AB) = \det A \cdot \det B$.

Steps

1. One transformation after another
2. Multiply
3. Does order matter?
4. Track a vector
5. Reflect twice
6. Determinants multiply