Interpreting $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \cdot h $ as a scalar or matrix multiplication As part of another problem I am working on, I have the following product to work out. 
$\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \cdot h $
where $h$ is a scalar. My question is, if I commute the row vector and the scalar then I can just multiply it through. If I think of the $h$ as a $1 \times 1$ matrix however, it seems that this isn't allowed. 
I know it sounds simple, but I want to understand the subtlety involved. 

There's not really a "subtlety" so to speak. Matrix products are not commutative. So $h\cdot A$ for a matrix $A$ is different than $A\cdot h$ for the same matrix $A$. In your example, $h$ is a scalar and may be thought of as a $1\times 1$ matrix but your matrix $A$ is $1\times 3$. So the product $h\cdot A$ is defined whereas $A\cdot h$ is undefined. 
The real problem for you it seems is in understanding when a matrix product is defined. Does my explanation above make sense?
