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

The book by George W. Hart mentioned in the comments is actually a quite good reference on the topic.
Furthermore, I also gave a talk on how a C++ library can be designed that is able to solve the topic for the general case of non-uniform physical units in vectors and matrices:
https://www.youtube.com/watch?v=J6H9CwzynoQ
