Suppose M is a symmetric invertible matrix, and let
J = (({{0, 1}, {-1, 0}})). Then

 	M^(-1)=((-J . M . J)/(Det (M)))

Proof:

M = {{A, B}, {B, CC}}

Inverse[M]

J = {{0, 1}, {-1, 0}}
J  //  MatrixForm

-1/Det[M] J . M . J

Inverse[M] - (-1/Det[M] J . M . J)