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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 $p$?

  1. $p^{2}-1$
  2. $p-1$
  3. $p$
  4. $p+1$
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