Row-by-column product

The product of matrices is not done element by element: it follows the row-by-column rule, and for this reason it requires the dimensions to be compatible.

Property — Row-by-column product

If $A$ is $m\times p$ and $B$ is $p\times n$, the product $C=AB$ is $m\times n$ with $c_{ij} = \sum_{k=1}^{p} a_{ik}\,b_{kj}.$ Mnemonic: “row $i$ of $A$ against column $j$ of $B$”. The dimensions must “glue together”: $(m\times p)(p\times n) = m\times n$. It is not commutative: in general $AB\ne BA$ even when both products are defined.

Connections

Topics: Linear systems
Concepts: Matrix · Product of matrices
Skills: Using formulae