Taylor Expansion with matrices

Question

The following formula should be expanded with the Taylor Expansion in $\lambda$: $$(A-\lambda B)^{-1}$$ In a first step this can be re-written as $$(\mathbf 1-\lambda A^{-1}B)^{-1}A^{-1}$$ The taylor expansion is then supposed to be $$\left(\mathbf 1+\sum_{n=1}^\infty(\lambda A^{-1}B)^n\right)A^{-1}$$ However, I don't understand how one arrives there - in fact, I don't even understand how the function is supposed to be differentiated.