---
tag: 09GK5PS824
kind: statement
role: proposition
name: Transpose of a product
msc: [15A99]
uses: [0W809K6AW0, 076JGP6R3T]
sources:
  - url: https://en.wikipedia.org/wiki/Transpose
    locator: Properties
---

## Statement

Let $R$ be a commutative ring.
Let $A$ be an $m \times n$ matrix and $B$ an $n \times p$ matrix over $R$.
Then $(AB)^T = B^T A^T$, where $^T$ denotes the [[transpose|09DXWYRD43]].

## Remarks

The order of the factors reverses.
The proof compares entries: both sides have entry $(k, i)$ equal to $\sum_{j} a_{ij} b_{jk}$.
By induction, $(A_1 A_2 \cdots A_k)^T = A_k^T \cdots A_2^T A_1^T$.
If $R$ is not commutative, the identity can fail.

## Explanations

### L1 human

The transpose of a product is the product of the transposes in the reverse order.
This is like taking off socks and shoes: the last thing put on comes off first.
The sizes also force this order, because $B^T$ is $p \times n$ and $A^T$ is $n \times m$.