Complex matrix inversion by Real matrix inversion

Calculate the complex matrix inverse by using only real matrix inverse

Given a complex square matrix M = A + i*B, its inverse is also a complex square matrix Z = X + i*Y, where A, B and X, Y are all real matrices. It is found that
M^-1 = Z or
(A + i*B)^-1 = (A + B*A^-1*B)^-1 - i*(B + A*B^-1*A)^-1
Provided that those matrices involved inversion must be nonsingular.