Multiplication of Matrices

Multiplying two matrices is less straightforward than adding or subtracting them, because it isn't simply a matter of multiplying corresponding elements. Two matrices $A$ and $B$ can be multiplied, in that order, only when the number of columns in $A$ equals the number of rows in $B$ — if $A$ is of order $m \times n$ and $B$ is of order $n \times p$, the product $AB$ exists and has order $m \times p$. Each element of the product is found by pairing a row of $A$ with a column of $B$ and combining them:

In words: to get the entry in row $i$, column $j$ of $AB$, multiply each element of row $i$ of $A$ with the corresponding element of column $j$ of $B$, and add up the results.