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}}

{{A, B}, {B, CC}}

Inverse[M]

{{CC/(-B^2 + A CC), -B/(-B^2 + A CC)}, {-B/(-B^2 + A CC), A/(-B^2 + A CC)}}

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

{{0, 1}, {-1, 0}}

( 0 1 ) -1 0

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

{{CC/(-B^2 + A CC), -B/(-B^2 + A CC)}, {-B/(-B^2 + A CC), A/(-B^2 + A CC)}}

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

{{0, 0}, {0, 0}}

Created by Mathematica  (March 10, 2006)