Matrix Multiplication — Mathematics Knowledge System

Overview

Matrix multiplication is a binary operation that produces a new matrix from two matrices. For the product AB to be defined, the number of columns of A must equal the number of rows of B: an m×n matrix times an n×p matrix yields an m×p matrix. Unlike ordinary number multiplication, matrix multiplication is not commutative in general — AB ≠ BA — because it represents the composition of two linear transformations, and the order in which transformations are applied matters.

Intuition

Each entry of the product AB is a dot product: the i-th row of A against the j-th column of B. A useful mental model is that matrix multiplication composes functions — if A and B are linear maps, AB means 'first apply B, then apply A' (reading right to left), just like function composition f∘g means 'first g, then f'. That's also why shapes must match up correctly and why order matters.

Interactive Graph

Formal Definition

Definition

For an m×n matrix A and an n×p matrix B, the product C = AB is the m×p matrix whose (i,j) entry is the dot product of row i of A with column j of B:

\mathbf{A} = (a_{ik})_{m \times n},\quad \mathbf{B} = (b_{kj})_{n \times p}

A is m×n, B is n×p — the inner dimensions (n) must match

c_{ij} = \sum_{k=1}^{n} a_{ik}\, b_{kj}

The (i,j) entry of C = AB: row i of A dotted with column j of B

Entry formula

\mathbf{C} = \mathbf{AB} \text{ is } m \times p

The resulting shape: rows of A by columns of B

Notation

Notation	Meaning
$\mathbf{AB}$	The matrix product of A and B (A applied after... reading conventions vary; algebraically, row-times-column)
$c_{ij}$	The (i,j) entry of the product matrix C = AB
$\mathbf{I}_n$	Identity matrix — AI = IA = A for compatible shapes

Properties

Associativity

(AB)C = A(BC)

Condition: Whenever the products are defined

Distributivity over addition

A(B+C) = AB + AC,\quad (A+B)C = AC+BC

NOT commutative in general

AB \neq BA

Example: Even when both products are defined (e.g. both square of the same size), equality is the exception, not the rule

Transpose of a product

(AB)^{\mathsf T} = B^{\mathsf T}A^{\mathsf T}

Condition: Order reverses under transpose

Identity element

A\, I_n = I_m\, A = A

Condition: For A of shape m×n

Applications

Composing sequences of transformations (e.g. successive graph adjacency-matrix powers count walks of a given length) and batch linear algebra in numerical libraries all reduce to matrix multiplication.

Worked Examples

c₁₁ = row 1 of A · column 1 of B = 1·5 + 2·7.
$c_{11} = 1(5) + 2(7) = 5 + 14 = 19$
c₁₂ = row 1 of A · column 2 of B = 1·6 + 2·8.
$c_{12} = 1(6) + 2(8) = 6 + 16 = 22$
c₂₁ = row 2 of A · column 1 of B = 3·5 + 4·7.
$c_{21} = 3(5) + 4(7) = 15 + 28 = 43$
c₂₂ = row 2 of A · column 2 of B = 3·6 + 4·8.
$c_{22} = 3(6) + 4(8) = 18 + 32 = 50$

Answer: AB = [[19, 22], [43, 50]]

Practice Problems

Difficulty 4/10

Compute AB for A = [[2, 0], [1, 3]] and B = [[4, 1], [0, 2]].

Difficulty 3/10

A is 3×5 and B is 5×2. What is the shape of AB, and is BA defined?

AB is 3×2; BA is not defined (2≠3 mismatch on inner dimension)AB is 5×5; BA is defined and equals ABAB is 3×2; BA is also 3×2Neither AB nor BA is defined

Difficulty 5/10

In 2-D computer graphics, a point (x, y) is rotated 90° counter-clockwise by multiplying by R = [[0, −1], [1, 0]]. Where does the point (3, 1) go?

Difficulty 6/10

In the stiffness method, member forces are found from f = k·d where k is the element stiffness matrix and d the nodal displacements. If k = [[50, −50], [−50, 50]] kN/mm and d = [2, 4]ᵀ mm, compute the nodal forces f.

Common Mistakes

Common Mistake

Multiplying matrices entrywise (like addition), i.e. computing cᵢⱼ = aᵢⱼ·bᵢⱼ.

Matrix multiplication is row-times-column (dot products), not entrywise. The entrywise product (Hadamard product) is a different, less common operation and must be explicitly named as such.

Common Mistake

Assuming AB = BA, as with ordinary numbers.

Matrix multiplication is generally non-commutative; swapping the order can even change whether the product is defined at all, since it depends on matching inner dimensions.

Quiz

Summary

For A (m×n) and B (n×p), the product AB is m×p, with entry cᵢⱼ = Σₖ aᵢₖ bₖⱼ.
The product is defined only when A's number of columns equals B's number of rows.
Matrix multiplication is associative and distributive over addition, but NOT commutative in general.
(AB)ᵀ = BᵀAᵀ — the order reverses under transpose.
Matrix multiplication models composition of linear transformations, chained geometric transforms, and neural network layers.

References

WebsiteWikipedia — Matrix multiplication
BookStrang, G. Introduction to Linear Algebra, 5th ed. Ch. 2.