\displaystyle $\begin{align}
& (A + N) \sum_{k = 0}^{p - 1} (-1)^k N^k A^{-k - 1} = \sum_{k = 0}^{p - 1} (-1)^k N^k A^{-k} + \sum_{k = 0}^{p - 2} (-1)^k N^{k + 1} A^{-k - 1} = \\
= & [l = k + 1] = \sum_{k = 0}^{p - 1} (-1)^k N^k A^{-k} + \sum_{l = 1}^{p - 1} (-1)^{l - 1} N^l A^{-l} = (-1)^0 N^0 A^0 = I
\end{align}$