Skill: Determine whether two matrices can be multiplied

To decide whether $AB$ is defined, compare the number of columns in $A$ with the number of rows in $B$. These are the inner dimensions: they must match. If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ has dimensions $m\times p$.

For example, let

$$A=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}, \qquad B=\begin{bmatrix}1&0\\2&1\\3&4\\5&6\end{bmatrix}.$$

• $A$ has dimensions $2\times 3$, and $B$ has dimensions $4\times 2$.
• For $AB$, the inner dimensions are $3$ and $4$. They do not match, so $AB$ is not defined.
• For $BA$, the inner dimensions are $2$ and $2$. They match, so $BA$ is defined and has dimensions $4\times 3$.

Always check the dimensions in the order the matrices are written: reversing the order can change whether a product is defined.