If $A$ is an $n\times m$ matrix and $B$ is an $m\times k$ matrix, then the product $AB$ is the $n\times k$ matrix whose $(i,j)$ entry is the [[row-column multiplication|row-column product]] of the $i$th row of $A$ with the $j$th column of $B$. That is:
\[ (AB)_{i,j} = \text{row}_i(A)\cdot \text{col}_j(B).\]

If we want to emphasize that we are multiplying matrices in this way, we might sometimes write $A\cdot B$ instead of $AB$.

If $A$ is an $n\times m$ matrix and $B$ is an $\ell\times k$ matrix with $m\ne \ell$, then the matrix product $AB$ is undefined.