Matrix Product

Definition

Let

• $A$ : $m\times r$ matrix
• $B$ : $r\times n$ matrix

Then the product $AB$ is a $m\times n$ matrix, whose entries are determined as follows:

1. To find the entry in row $i$ and column $j$ of $AB$, single out row $i$ from the matrix $A$ and column $j$ from the matrix $B$.
2. Multiply the corresponding entries from the row and column together, and then add up the resulting products.

Remark

See matrix multiplication for example. As shown in the example, we can obtain specific entry $(AB{)}_{ij}$ only by knowing the $i$-th row of $A$ and $j$-th row of $B$:

$$(AB{)}_{ij}=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+\cdots +a_{ir}{b}_{rj}$$

This formula is called the row-column rule for matrix multiplication.

One easy trick to determine the size of matrix multiplication is by looking at the pattern

$$\begin{array}{cccc}A& B& & AB\\ m\times r& r\times n& =& m\times n\end{array}$$

The “inner” size $r$ has to be equal for the multiplication to be defined.