Matrix multiplication is a grid of dot products

One output entry is one dot product

Think of a row as a rule and a column as an input. To apply that rule, multiply matching entries and add. A matrix product performs all the compatible row–column pairings at once.

For example,

\begin{bmatrix}1&2\\3&4\end{bmatrix} \begin{bmatrix}5\\6\end{bmatrix} =\begin{bmatrix}1\times5+2\times6\\3\times5+4\times6\end{bmatrix} =\begin{bmatrix}17\\39\end{bmatrix}.

The first row produces the first output. The second row produces the second. Both read the same two input coordinates.

The shapes tell a story

If $A$ has shape $m\times n$ and $B$ has shape $n\times p$ , then

\underbrace{A}_{m\times n} \underbrace{B}_{n\times p} =\underbrace{AB}_{m\times p}.

The shared $n$ is the number of terms in each dot product. The output has one row for each rule in $A$ and one column for each input in $B$ .

In symbols,

(AB)_{ij}=\sum_{k=1}^{n}A_{ik}B_{kj}.

The index $k$ travels along a row of $A$ and down a column of $B$ . The indices $i,j$ tell us where to put the answer.

Try a second input

What happens if we add the input column $(0,1)$ beside $(5,6)$ ?

Answer: We get a second output column, $(2,4)$ . The same matrix applies its rules to each input independently.