Matrix Multiplication

A precise explanation of matrix multiplication as composition, row–column pairing, and column combination.

A matrix-product entry is one row–column dot product. The first row of a two-by-two matrix and the first column of a second matrix are highlighted. Their paired products add to the highlighted top-left entry of the result matrix.

Precise definition

If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$ with $(AB)_{ij}=\sum_{k=1}^n a_{ik}b_{kj}$. The inner dimensions must match. Multiplication is associative and distributive but generally not commutative.

Notation and mathematical language

Column $j$ of $AB$ is $A$ times column $j$ of $B$, hence a linear combination of $A$'s columns. As transformations, $AB$ represents applying $B$ first and then $A$ under compatible coordinate bases.

Conceptual picture

The row–column formula measures how an output coordinate of $A$ combines every intermediate coordinate supplied by $B$. This contraction of the shared index explains both the dimension rule and the order of composition.

Fully worked example

Interpretation and application

Matrix products combine transformations, transition steps, network paths, and linear models. The order encodes process order and therefore carries substantive meaning rather than being a formatting choice.