Multiplication of Matrices

Conformability for multiplication

Two matrices $A$ (order $m\times n$) and $B$ (order $p\times q$) can be multiplied, as $AB$, only if $n=p$ — i.e. the number of COLUMNS of $A$ equals the number of ROWS of $B$. The product $AB$ then has order $m\times q$. For example, a $2\times3$ matrix can be multiplied by a $3\times2$ matrix (giving a $2\times2$ product), but not by another $2\times3$ matrix.

The row-by-column rule

Each element of $AB$ is found by taking a ROW of $A$ and a COLUMN of $B$, multiplying corresponding entries together, and adding the results. For $2\times2$ matrices,

The element in row $i$, column $j$ of $AB$ always comes from ROW $i$ of $A$ combined with COLUMN $j$ of $B$.

Multiplication is NOT commutative

Unlike ordinary numbers, matrix multiplication generally does not commute: $AB$ and $BA$ are usually DIFFERENT matrices — and one product may not even be defined when the other is (e.g. for a $2\times3$ matrix times a $3\times2$ matrix, $AB$ and $BA$ are both defined but have different orders, $2\times2$ and $3\times3$ respectively, so they cannot possibly be equal).

The identity matrix under multiplication

For any square matrix $A$ and the identity matrix $I$ of the same order, $AI=IA=A$ — multiplying by the identity matrix, in either order, always leaves a matrix unchanged. This is the one case where the order of multiplication makes no difference. …