Symbolics Writing element (matrix) inverse


I am trying to create a symbolic representation of some algebra that I am trying to work out for a project. Ideally, I would like to create a way to write an expression like below:

@variables H, \sigma # H is an nxn matrix, \sigma is a number
A = [H 0; 0 \sigma] # where H is "like" a block matrix 

Then, inv(A) becomes [1/H 0; 0 1 / \sigma]
Instead of 1/H, I want this to read as H^{-1}. Is this possible to do symbolically?

Alternatively, is there a way to have a @variables where the variable we create is not necessarily considered a Num, but instead a matrix of size n x n, but I do not need to specify the exact size of n? I think this would then allow the behavior I am hoping for above.

Thank you!