It's not true that each element can have its own units; given an MxN matrix there are only M+N-1 degrees of freedom (basically you can choose arbitrary units fo

It's not true that each element can have its own units; given an MxN matrix there are only M+N-1 degrees of freedom (basically you can choose arbitrary units for each element in the first row and then a single multiplicative factor between the first row and each other row). Otherwise you'll find that either left or right multiplication won't have any compatible vector types.