Let $p$ be a prime number, and let $A$ equal $\left(\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right)$, viewed as a $2 \times 2$ matrix with integer entries. What is the smallest positive integer $n$ such that the matrix $A^{n}$ is congruent to the $2 \times 2$ identity matrix modulo $p$?
- $p^{2}-1$
- $p-1$
- $p$
- $p+1$