Q.Do as directed:

Matrix Multiplication: Why It Works the Way It Does

You already know how to multiply numbers. $3 \times 5 = 15$. Simple. Matrix multiplication looks different — and it is different — but there's a reason for every rule.

The Intuition: A Factory Analogy

Imagine you run a factory that makes two products: chairs and tables. Each product needs raw materials: wood and metal.

Let the first matrix tell you how much of each material goes into each product:

This is a $2 \times 2$ matrix: $A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$.

Now suppose you have two different price lists for wood and metal — one from Supplier X, one from Supplier Y:

This is a $2 \times 2$ matrix: $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$.

You want to know: What is the total cost of making one chair using Supplier X's prices? You take the wood cost ($2 \times 5$) plus the metal cost ($1 \times 7$) = $10 + 7 = 17$.

That single number — 17 — is the first entry of the product matrix $AB$. It comes from the first row of $A$ (chair's material needs) dotted with the first column of $B$ (Supplier X's prices).

The Precise Definition

If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then their product $C = AB$ is an $m \times p$ matrix where:

That sum is just a compact way of saying: multiply each element in row $i$ of $A$ by the corresponding element in column $j$ of $B$, then add them all up.

Properties That Hold (and One That Doesn't)

1. Associativity: $(AB)C = A(BC)$ — as long as the dimensions line up, the grouping doesn't matter. This is a lifesaver in calculations.

2. Distributivity: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$ — exactly like numbers.

3. Identity: There is an identity matrix $I$ (1's on the diagonal, 0's elsewhere) such that $AI = A$ and $IA = A$.

Why Commutativity Fails …