Matrix inverse series expansion

Question

I want to prove that when $I+K$ is invertible, $$(I+K)^{-1}=I-K+o(K)$$ to establish that the matrix inverse function has derivative $-I$ at $I$. My hope is that this identity carries over from $\mathbb{C}$, where we have $$(1+z)^{-1}=1-z+o(z)$$ and while we obviously have the expansion $(I-K)(I+K)=I-K^{2}=I+o(K)$, I'm not sure if this can be used to establish the above.