Multiplication of Matrices

Matrix multiplication models a very natural business operation: combining a quantity matrix with a price/rate matrix to produce a total matrix (as in the worked revenue example below), so it is worth understanding the mechanics precisely.

The Conformability Rule

Two matrices $A$ (order $m\times n$) and $B$ (order $p\times q$) can be multiplied, in the order $AB$, only if the number of columns of $A$ equals the number of rows of $B$, i.e. $n=p$. The product $AB$ then has order $m\times q$ — the "outer" dimensions.

How to Multiply

Each element of $AB$ is obtained by multiplying a row of $A$ with a column of $B$, entry by entry, and adding:

that is, the element in row $i$, column $j$ of $AB$ is the sum of the products of the $i$-th row of $A$ with the $j$-th column of $B$.

For $2\times 2$ matrices $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ and $B=\begin{pmatrix}e&f\\g&h\end{pmatrix}$:

The same row-into-column pattern extends to $3\times 3$ matrices, with three terms summed in each entry instead of two.

Non-Commutativity …