Multiplication of Matrices

The Compatibility Condition

Unlike addition, matrix multiplication does not require the two matrices to have the same order -- but it does require a different kind of match.

Definition. If $A$ is a matrix of order $m\times n$ and $B$ is a matrix of order $n\times p$ -- that is, the number of columns of $A$ equals the number of rows of $B$ -- then the product $AB$ is defined, and is a matrix of order $m\times p$. If the column-count of $A$ does not equal the row-count of $B$, the product $AB$ is simply not defined.

The Row-by-Column Rule

Each entry of the product is computed by pairing a full row of $A$ with a full column of $B$:

In words: to find the entry in row $i$, column $j$ of $AB$, take row $i$ of $A$, take column $j$ of $B$, multiply them term by term, and add up the results. Example 5 and Exercise: Matrix Multiplication, Q1 both apply this rule directly for $2\times2$ matrices:

The same rule extends to rectangular matrices exactly as stated in the compatibility condition above -- Exercise: Matrix Multiplication, Q3 multiplies a $2\times3$ matrix by a $3\times2$ matrix to get a $2\times2$ product, working through each entry as a sum of three products (since $n=3$ here) rather than two.

Properties That Multiplication Does Satisfy

Although matrix multiplication behaves very differently from ordinary multiplication in some respects (Section 8 covers exactly where it differs), it does satisfy the following, whenever the products involved are defined:

• Associativity: $(AB)C=A(BC)$.
• Distributivity over addition: $A(B+C)=AB+AC$ and $(A+B)C=AC+BC$.
• Compatibility with scalars: $k(AB)=(kA)B=A(kB)$ for any scalar $k$. …