SIMON-MARAIS 2025 A2

Let $p$ be a prime number. Given a square matrix $M$ with entries in $\mathbb{Z}/p\mathbb{Z}$, an eigenvector of $M$ is a non-zero vector $\mathbf{v}$ with entries in $\mathbb{Z}/p\mathbb{Z}$ such that $M\mathbf{v} = \lambda\mathbf{v}$ for some $\lambda \in \mathbb{Z}/p\mathbb{Z}$.

For each positive integer $n$, determine the average number of eigenvectors of an $n \times n$ matrix with entries in $\mathbb{Z}/p\mathbb{Z}$.

(Here $\mathbb{Z}/p\mathbb{Z}$ refers to the integers modulo $p$.)