Let $A \in M_{n \times n}(\mathbb{C})$.\\
(a) Suppose that $A^{2} = 0$. Show that $\lambda$ is an eigenvalue of $(I_{n}+A)$ if and only if $\lambda = 1$. ($I_{n}$ is the $n \times n$ identity matrix.)\\
(b) Suppose that $A^{2} = -1$. Determine (with proof) whether $A$ is diagonalizable.