Let $A$ be an $n \times n$ matrix such that $A^k = 0_{n,n}$ (the $n \times n$ zero matrix)
for some natural integer $k$. Show that I $_{n} + A$ is invertible.
2. $(I_n+A)(I_n-A+A^2-\ldots+(-1)^{k-1}A^{k-1})=\ldots=I_n$ .