---
tag: 05Q1B7MH0M
kind: statement
role: theorem
name: Matrix of a composition is the product of matrices
msc: [15A04]
uses: [0YYMY95FWS, 05YBKMRVAR, 076JGP6R3T]
sources:
  - url: https://en.wikipedia.org/wiki/Transformation_matrix
    locator: Composing and inverting transformations
---

## Statement

Let $F$ be a field, and let $U$, $V$, $W$ be finite-dimensional vector spaces over $F$ with bases $A$, $B$, $C$.
Let $S \colon U \to V$ and $T \colon V \to W$ be linear maps.
Then
$$[T \circ S]_{C,A} = [T]_{C,B}\, [S]_{B,A},$$
where $[\,\cdot\,]$ denotes the [[matrix of a linear map|0YQSGNYE3K]] and the right side is a [[matrix product|032Q9FB8MT]].

## Remarks

This identity is the reason for the definition of matrix multiplication.
It implies that matrix multiplication is associative, because composition of maps is associative.
For $S$ invertible, it gives $[S^{-1}]_{A,B} = [S]_{B,A}^{-1}$.

## Explanations

### L1 human

Doing one linear map after another corresponds to multiplying their matrices.
The order is the same as in $T \circ S$: the matrix of the map that acts first is on the right.
This is why matrix multiplication has its unusual row-times-column rule.