Matrix multiplication

Definition

Let $R$ be a ring, let $A = (a_{ij})$ be an $m \times n$ matrix over $R$, and let $B = (b_{jk})$ be an $n \times p$ matrix over $R$. The product $AB$ is the $m \times p$ matrix with entries $$(AB)_{ik} = \sum_{j=1}^{n} a_{ij} b_{jk}$$ for $1 \le i \le m$ and $1 \le k \le p$.

Remarks

The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$. Column $k$ of $AB$ is the matrix–vector product of $A$ with column $k$ of $B$. The product is associative and distributive, but it is not commutative. If $A$ and $B$ are the matrices of two linear maps, then $AB$ is the matrix of their composition.

Explanations

L1 human

To find the entry in row $i$ and column $k$ of $AB$, go along row $i$ of $A$ and down column $k$ of $B$. Multiply the pairs of numbers and add the products. For example, $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 11 \\ 4 \end{pmatrix}$.