Let $B$ be a square matrix with coefficients in $\mathbb{C}$ such that $B^3 = 0$. What is the necessary and sufficient condition on $\alpha$ for the matrix $B + \alpha I_n$ to be invertible? Provide the inverse matrix in that case.
I attempted to calculate $(B + \alpha I)^m$ for $m \in \mathbb{N}$ using the binomial theorem (retaining only terms with a power of $k$ less than 3), but it did not yield any meaningful result.