I want to understand how this expansion of a matrix to this inverse of partitioned matrix form is done.

Question

I see people making use of this technique to represent 4 energy bands problems to 2 bands problems in condensed matter physics (I'm looking at bilayer graphene particularly).

original matrix in its partitioned form_eq.1

matrix expanded_eq.2

a) I wish to understand how we go from this original matrix to the expanded form of the matrix.

b) And also, how this second equation makes sense.

c) And how representing 4 band hamiltonian matrix in an effective 2 band hamiltonian matrix make sense.