Matrix multiplication — definition, associativity, non-commutativity: Foundations

Before you can read the parent note, you need to own a handful of symbols. This page builds each one from nothing: plain words first, then a picture, then why the topic can't live without it. We go in dependency order — each idea leans only on the ones above it.

1. Numbers in a box — what a matrix even is

Plain words: a matrix is just a table of numbers, nothing more mysterious than a spreadsheet. The magic is what we later do with it.

The picture below shows the same grid with its rows and columns coloured — burn this layout into your eyes, because every later rule is "grab a row, grab a column".

Why the topic needs it: the whole parent note is about combining these grids. If you can't point at "row 2" and "column 1" instantly, the row·column rule will feel like magic instead of geography.

2. Naming a single number inside — the subscript $A_{ij}$

For the matrix $A = (1324)$ :

$A_{11} = 1$ (row 1, column 1),
$A_{12} = 2$ (row 1, column 2),
$A_{21} = 3$ (row 2, column 1),
$A_{22} = 4$ (row 2, column 2).

Why the topic needs it: the master formula $C_{ij} = \sum_{k} A_{ik} B_{k j}$ is entirely made of these addresses. If $A_{ij}$ is fuzzy, that formula is unreadable.

3. The two shape numbers — $m \times n$

$(142536)$ is $2 \times 3$ : two rows, three columns.
A single column $(57)$ is $2 \times 1$ .

Why the topic needs it: shape mismatch is the #1 reason a product doesn't exist, and unequal shapes are reason 1 the parent gives for non-commutativity ( $A B$ can be $2 \times 2$ while $B A$ is $3 \times 3$ ).

4. A vector — the thing a matrix acts on

Here $x_{1}$ and $x_{2}$ are the vector's components — its individual coordinate numbers. The subscript is now just position-in-the-list (there's only one column, so no need for a second index).

Why the topic needs it: the parent defines the product by demanding $(A B) x = A (B x)$ for every vector $x$ . Without knowing $x$ is a grabbable arrow, the phrase "matrices are machines that move vectors" has nothing to move. See Linear Transformations for the full "machine" story.

5. The dot product — one number from two lists

Worked in the flesh: $(1, 0, 2) \cdot (4, 0, 1) = 1 \cdot 4 + 0 \cdot 0 + 2 \cdot 1 = 6$ .

Why the topic needs it: the parent's headline is " $(A B)_{ij} =$ row $i$ of $A$ dotted with column $j$ of $B$ ." Every single entry of every product is one dot product. See Dot Product for its geometric meaning (length and angle).

6. The summation sign $\sum$ — shorthand for "add these up"

Plain words: $\sum$ is just a loop that totals things. The letter under it ( $k$ here) is a throwaway counter — it appears only inside and vanishes from the answer. Notice this is literally the dot product from §5 written with a loop instead of " $+ \dots +$ ".

Why the topic needs it: the entire definition, the associativity proof, and the double sums $\sum_{k} \sum_{j}$ are written in $\sum$ . It's the language the whole page speaks.

7. Special matrices you'll meet

8. Symbols that appear later (so nothing surprises you)

You don't need to compute these yet — just recognise the symbols when the parent uses them. Powers like $A^{2} = AA$ and diagonalization live in Matrix Powers and Diagonalization.

How these feed the topic

Read it top-down: the plain grid gives you addresses and sizes; addresses + sizes + the sum sign build the dot product; the dot product plus "a matrix moves a vector" build the product rule; and the product rule branches into the two big theorems — associativity and non-commutativity — of the parent topic.

Equipment checklist

Cover the right side and test yourself — if any answer is fuzzy, reread its section.

A_{ij}

, which subscript is the row?

The first one,

i

; the second

j

is the column (Row then Column).

What does "

A

3 \times 2

" tell you?

A

has 3 rows and 2 columns (rows first, columns second).

Compute

(2, 1, 0) \cdot (3, 4, 5)

2 \cdot 3 + 1 \cdot 4 + 0 \cdot 5 = 10

— multiply matching entries, add up.

What does

\sum_{k = 1}^{2} A_{ik} B_{k j}

expand to?

A_{i 1} B_{1 j} + A_{i 2} B_{2 j}

— loop

k

over

1, 2

and add.

What is a vector, geometrically?

A column of numbers = an arrow from the origin to the point with those coordinates.

Which dimensions must match to multiply

A

(size

m \times n

) by

B

The inner ones:

B

must be

n \times p

; result is

m \times p

What does the identity matrix

I

do to any vector?

Nothing —

I x = x

; it's the do-nothing machine.

What does the commutator

[A, B]

equal, and what does zero mean?

[A, B] = A B - B A

; it is

0

exactly when

A

and

B

commute.

Why is

I

called the reference for commuting?

Every matrix commutes with

I

(

A I = I A = A

), so it's the baseline "always-commutes" case.

Connections

Matrix multiplication — definition, associativity, non-commutativity — the parent this page equips you for.
Linear Transformations — the "matrix = machine that moves vectors" picture in full.
Dot Product — the multiply-and-add operation each entry uses.
Identity and Inverse Matrices — $I$ , $A^{- 1}$ , and why order reverses.
Transpose — the flip symbol $A^{⊤}$ .
Determinants — the stretch-factor number $det A$ .
Matrix Powers and Diagonalization — powers $A^{2}$ and commuting matrices.