$A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$

$A^2=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}3 & -3 \\ 3 & 0\end{array}\right]$

$A^3=\left[\begin{array}{cc}3 & -3 \\ 3 & 0\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}3 & -6 \\ 6 & -3\end{array}\right]$

$A^4=\left[\begin{array}{ll}3 & -6 \\ 6 & -3\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}0 & -9 \\ 9 & -9\end{array}\right]$

$A^5=\left[\begin{array}{ll}0 & -9 \\ 9 & -9\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}-9 & -9 \\ 9 & -18\end{array}\right]$

$A^6=\left[\begin{array}{cc}-9 & -9 \\ 9 & -18\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}-27 & 0 \\ 0 & -27\end{array}\right]$

$A^7=\left[\begin{array}{cc}-27 & -0 \\ 0 & -27\end{array}\right]\left[\begin{array}{cc}-54 & 27 \\ -27 & -27\end{array}\right]=\left[\begin{array}{cc}3^6 \times 2 & -27^2 \\ 27^2 & 3^6\end{array}\right]$

$3^7=3^{\mathrm{n}} \Rightarrow \mathrm{n}=7$