Transpose of a product

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.

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 $n \times m$.