[porges]

Each element of your matrix doesn't really have its own type, each column and row has a type:

                            1
                          +---
                       V  | V
                      Pa  | Pa

          V^-1    Pa^-1   
         +--------------- +-----
    mA   | mA/V   mA/Pa   | mA
    kg/s | kg/V/s kg/Pa/s | kg/s

I can imagine a system that types this quite well.