(({{1, b (d^(-1))}, {0, 1}})).(({{a-b (d^(-1))c, 0}, {0, d}})).(({{1, 0}, {(d^(-1))c, 1}}))

Simplify[Inverse[(({{a, b}, {c, d}}))]== (({{(((a-b (d^(-1))c))^(-1)), -(((a-b (d^(-1))c))^(-1))b (d^(-1))}, {-(d^(-1))c(((a-b (d^(-1))c))^(-1)), (d^(-1))+(d^(-1))c(((a-b (d^(-1))c))^(-1))b (d^(-1))}}))]

Simplify[(d^(-1))+(d^(-1))c(((a-b (d^(-1))c))^(-1))b (d^(-1))]

(({{1, b λ (d^(-1))}, {0, 1}})).(({{a-b λ (d^(-1))c, 0}, {0, d}})).(({{1, 0}, {λ (d^(-1))c, 1}})) // Simplify // MatrixForm

(({{1, b λ (d^(-1))}, {0, 1}})).(({{a-b λ (d^(-1))c, 0}, {0, (λ^(-1))d}})).(({{1, 0}, {λ (d^(-1))c, 1}})) // Simplify // MatrixForm

A=Table[(Subscript[a, i,j]),{i,4},{j,4}]

Minors[Det[A]Inverse[A],2]/Det[A] // Together

p=3; q=2;
n=p+q;
A=Table[(Subscript[α, 10i+j]),{i,n},{j,n}];
Map[MatrixForm,(({{a=A[[1;;p,1;;p]], b=A[[1;;p,p+1;;n]]}, {c=A[[p+1;;n,1;;p]], d=A[[p+1;;n,p+1;;n]]}})),{2}]//MatrixForm

Simplify[Det[A]*Inverse[a-b.Inverse[d].c]]//MatrixForm

Ai=Simplify[Inverse[A]];
di=Simplify[Inverse[d]];
api=Simplify[Inverse[a-b.di.c]];
Simplify[(({{Ai[[1;;p,1;;p]], Ai[[1;;p,p+1;;n]]}, {Ai[[p+1;;n,1;;p]], Ai[[p+1;;n,p+1;;n]]}}))==(({{api, -api.b.di}, {-di.c.api, di+di.c.api.b.di}}))]