Symbolically computition of G'=ZG+δI

I want to calculate the Sign function of a matrix Z by using the Green's matrix function G[t].The above steps will help understanding the problem: 1]Solve G'[t]=ZG[t]+δΙ Where δ is the Dirac function,I the identity matrix,Z the constant matrix and G[t] which i want. 2] We take the limits limit t->0+G[t]=G[+0] and t->0-G[t]=G[-0] 3]the SignZ=-(G[+0] +G[-0] ). I have already do these steps but one property for SignZ is that (signZ)^2=I. Any opinion ? With Best Regards Dimitrios