Let A= [ … ]. If the sum of the diagonal elements of A^13 is 3^n, then n is equal to

$A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$A=[21​−11​] $\begin{aligned} & A^2=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}3 & -3 \\ 3 & 0\end{array}\right] \\ & A^3=\left[\begin{array}{cc}3 & -3 \\ 3 & 0\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}3 & -6 \\ 6 & -3\end{array}\right] \\ & A^4=\left[\begin{array}{cc}3 & -6 \\ 6 & -3\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}0 & -9 \\ 9 & -9\end{array}\right] \\ & A^5=\left[\begin{array}{cc}0 & -9 \\ 9 & -9\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}-9 & -9 \\ 9 & -18\end{array}\right] \\ & A^6=\left[\begin{array}{cc}-9 & -9 \\ 9 & -18\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}-27 & 0 \\ 0 & -27\end{array}\right] \\ & A^7=\left[\begin{array}{cc}-27 & -0 \\ 0 & -27\end{array}\right]\left[\begin{array}{cc}-54 & 27 \\ -27 & -27\end{array}\right]=\left[\begin{array}{cc}3^6 \times 2 & -27^2 \\ 27^2 & 3^6\end{array}\right] \\ & 3^7=3^n \Rightarrow n=7\end{aligned}$​A2=[21​−11​][21​−11​]=[33​−30​]A3=[33​−30​][21​−11​]=[36​−6−3​]A4=[36​−6−3​][21​−11​]=[09​−9−9​]A5=[09​−9−9​][21​−11​]=[−99​−9−18​]A6=[−99​−9−18​][21​−11​]=[−270​0−27​]A7=[−270​−0−27​][−54−27​27−27​]=[36×2272​−27236​]37=3n⇒n=7​