Matrix multiplication — conditions, process, non-commutativity

Overview

Matrix multiplication is not element-wise multiplication. It's a composition of linear transformations, where each entry in the product is a dot product of a row and column. This operation has strict dimension requirements and, critically, order matters.

[!intuition] Why Matrix Multiplication Works This Way

Think of matrices as transformation machines. When you multiply $AB$, you're asking: "What happens if I first apply transformation $B$, then transformation $A$?"

WHY this definition? Because it preserves the composition of functions. If $B$ transforms vector $\vec{x}$ into $B\vec{x}$, and $A$ transforms that result into $A(B\vec{x})$, we want a single matrix $C = AB$ such that $C\vec{x} = A(B\vec{x})$.

The row-column dot product emerges because:

• Each column of $B$ tells you where a basis vector goes
• Each row of $A$ tells you how to combine those transformed vectors
• The $(i,j)$ entry of $AB$ is "how much of the $j$-th column of $B$ contributes to the $i$-th row of $A$'s output"

[!definition] Matrix Multiplication

For matrices $A_{m \times n}$ and $B_{n \times p}$, the product $C = AB$ is defined if and only if the ==number of columns in $A$ equals the number of rows in $B$==.

The resulting matrix $C$ has dimensions $m \times p$, where:

In words: The $(i,j)$ entry of $C$ is the ==dot product of the $i$-th row of $A$ with the $j$-th column of $B$==.

[!formula] Step-by-Step Process

Condition Check:

The inner dimensions (both $n$) must match. The outer dimensions ($m$ and $p$) give the result's shape.

Computation:

For each entry $C_{ij}$:

1. Take row $i$ of $A$: $[A_{i1}, A_{i2}, \ldots, A_{in}]$
2. Take column $j$ of $B$: $\begin{bmatrix} B_{1j} \\ B_{2j} \\ \vdots \ B_{nj} \end{bmatrix}$
3. Compute: $C_{ij} = A_{i1}B_{1j} + A_{i2}B_{2j} + \cdots + A_{in}B_{nj}$

WHY this works: Each term $A_{ik}B_{kj}$ represents "how much of the $k$-th intermediate dimension contributes through $A$'s row and $B$'s column."

[!example] Example 1 — Valid Multiplication

Check: $A$ is $2 \times \color{red}3$, $B$ is $\color{red}3 \times 2$ → inner dimensions match ✓
Result: $C$ will be $2 \times 2$

Compute $C_{11}$ (row 1 of $A$ · column 1 of $B$):

Why this step? We're summing contributions from each of the 3 intermediate dimensions.

Compute $C_{12}$ (row 1 of $A$ · column 2 of $B$):

Compute $C_{21}$ (row 2 of $A$ · column 1 of $B$):

Compute $C_{22}$ (row 2 of $A$ · column 2 of $B$):

Final result:

[!example] Example 2 — Demonstrating Non-Commutativity

Compute $AB$:

Compute $BA$:

Result: $AB \neq BA$ ✓

WHY? Because composition of transformations depends on order. Rotating then scaling is different from scaling then rotating.

[!example] Example 3 — Undefined Multiplication

Check: $A$ is $2 \times \color{red}2$, $B$ is $\color{red}3 \times 1$ → inner dimensions DON'T match (2 ≠ 3) ✗

Result: $AB$ is undefined. You cannot multiply these matrices.

Why? There's a dimensional mismatch—$A$'s rows expect 2 components, but $B$ provides 3.

[!mistake] Common Mistakes

Mistake 1: Thinking $AB = BA$

Why it feels right: Regular multiplication of numbers is commutative, so students assume matrices work the same way.

The fix: Matrices represent transformations, and order matters in transformations. Putting on socks then shoes≠ putting on shoes then socks. Even when both $AB$ and $BA$ are defined (square matrices of same size), they're usually different.

Verification: Always compute both and compare. Special cases exist (like $A = I$ or diagonal matrices with same diagonal), but they're exceptions.

Mistake 2: Multiplying wrong dimensions

Why it feels right: Students focus on "both are matrices" and forget the column-row matching rule.

The fix: Always write dimensions explicitly: $A_{m \times \color{red}n} \times B_{\color{red}n \times p}$. The red numbers must match. If they don't, stop—multiplication is undefined.

Mistake 3: Element-wise multiplication

Why it feels right: It seems simpler to just multiply $A_{ij} \times B_{ij}$ (called Hadamard product, notation $A \circ B$).

The fix: Matrix multiplication is row-column dot products, not entry-by-entry. The Hadamard product is a different operation entirely and requires same dimensions. Standard multiplication requires the column-row match.

[!recall]- Feynman Explanation (ELI12)

Imagine you have a list of recipes (matrix $B$) that tells you how to make smoothies. Each recipe says "use 2 banas, 1 apple, 3 strawberries."

Now you have a price list (matrix $A$) that tells you the cost per fruit in different stores.

When you multiply $A \times B$, you're answering: "What's the total cost of each smoothie recipe in each store?"

To find the cost of smoothie #1 in store #1, you:

• Take store #1's prices (row 1 of $A$)
• Take smoothie #1's recipe (column 1 of $B$)
• Multiply each price by how many of that fruit you need
• Add them all up

That's one entry in your answer matrix!

Why can't you flip the order? Because "store prices × smoothie recipes" makes sense, but "smoothie recipes × store prices" doesn't—you can't multiply "2 bananas" by "$3 per apple" in a meaningful way. The inner numbers have to represent the same thing (fruits, in this case).

[!mnemonic] Memory Hook

"Columns meet Rows to make Entries"

• Columns of first matrix
• Rows of second matrix
• Must have same count
• Each pair makes an Entry via dot product

Alternatively: "ColRow →RowCol flips the result"
$AB \neq BA$ because you're composing transformations in opposite order.

Properties of Matrix Multiplication

1. Associative: $(AB)C = A(BC)$ ✓
2. Distributive: $A(B + C) = AB + AC$ ✓
3. NOT Commutative: $AB \neq BA$ in general ✗
4. Identity: $AI = IA = A$ where $I$ is the identity matrix ✓
5. Zero property: $A \cdot O = O$ where $O$ is the zero matrix ✓

WHY associative? Because function composition is associative: $(f \circ g) \circ h = f \circ (g \circ h)$.

Connections

• Matrix addition and scalar multiplication — simpler operations with same dimensions
• Identity matrix — the multiplicative "1" for matrices
• Inverse of a matrix — solving $AB = I$ to find $B = A^{-1}$
• Determinants — $\det(AB) = \det(A) \cdot \det(B)$ (comutativity holds for determinants!)
• Linear transformations — matrices as function representations
• Matrix transpose properties — $(AB)^T = B^T A^T$ (order reverses!)
• System of linear equations — $A\vec{x} = \vec{b}$ uses matrix-vector multiplication

Flashcards

What is the dimension requirement for multiplying $A_{m \times n}$ and $B_{p \times q}$? :: The number of columns in $A$ must equal the number of rows in $B$, i.e., $n = p$. Result is $m \times q$.

Concept Map

Hinglish (regional understanding)