matrix multiplication

$(AB)_{ij}$ is row $i$ of $A$ dotted with column $j$ of $B$ .

For $A$ an $n\times k$ matrix and $B$ a $k\times m$ matrix, the product $AB$ is the $n\times m$ matrix whose $(i,j)$ entry is $(AB)_{ij}=\sum_{t=1}^k a_{it}b_{tj}$ — the dot product of row $i$ of $A$ with column $j$ of $B$ . This is only defined when $A$ 's column count matches $B$ 's row count. Equivalently, column $j$ of $AB$ is $A$ times column $j$ of $B$ : each column of the product is a linear combination of $A$ 's columns.