To show that $\|A + I\| > 1$ for a nilpotent matrix $A$, let's proceed with the proof step by step.

Definition and Properties

1. Nilpotent Matrix: A square matrix $A$ is called nilpotent if there exists some positive integer $k$ such that $A^k = 0$. The smallest such $k$ is called the index of nilpotency.

2. Norm of a Matrix: There are different types of norms for matrices, but commonly used norms include the spectral norm (induced by the Euclidean norm on $\mathbb{R}^n$), the Frobenius norm, and others. For the sake of this proof, we will use a norm that is sub-multiplicative and satisfies $\|I\| = 1$.

Proof Outline

1. Nilpotency of $A$: Since $A$ is nilpotent, there exists an integer (m >