When should matrices have units of measurement? As a mathematician I think of matrices as $\mathbb{F}^{m\times n}$, where $\mathbb{F}$ is a field and usually $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$. Units are not necessary.
However, some engineers have told me that they prefer matrices to have units such as metres, kg, pounds, dollars, etc. Assigning a unit of measurement to each entry to me seems restrictive (for instance if working with dollars then is $A^2$ allowed?).
Here are a few things that I would like to understand more deeply:


*

*Are there examples where it is more appropriate to work with matrices that have units?

*If units can only restrict the algebra, why should one assign units at all?

*Is there anything exciting here, or is it just engineers trying to put physical interpretations on to matrix algebra?   
Also, see: 
https://stackoverflow.com/questions/11975658/how-do-units-flow-through-matrix-operations

Think about $Ax=b$ (not solving it, just the equation itself). If $x_j$ and $b_i$ have units, then $a_{ij} x_j$ has the units of $b_i$. Hence $a_{ij}$ has the units of $b_i/x_j$. So if $x$ is a vector of times and $b$ is a vector of positions then the entries of $A$ are velocities (or at least have those units).
I do not really understand how to interpret the idea that the matrix itself has units, however.
