Prove that if matrix $A$ is nilpotent, then $I+A$ is invertible. So my friend and I are working on this and here is what we have so far.
We want to show that $\exists \, B$ s.t. $(I+A)B = I$. We considered the fact that $I - A^k = I$ for some positive $k$. Now, if $B = (I-A+A^2-A^3+ \cdots -A^{k-1})$, then $(I+A)B = I-A^k = I$. My question is: in matrix $B$, why is the sign for $A^{k-1}$ negative? Couldn't it be positive, in which case we'd get $(I+A)B = I + A^k$?
Thank you.

It's the usual polynomial identity
$$
1 - x^{k} = (1 - x)(1 + x + x^{2} + \dots + x^{k-1}),
$$
where you are substituting $x = -A$.
