Let $$A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then $n$ is equal to ________.

<p>$$\begin{aligned} & A=\left[\begin{array}{cc} 2 & -1 \\ 1 & 1 \end{array}\right] \\ & A^2=\left[\begin{array}{cc} 2 & -1 \\ 1 & 1 \end{array}\right]\left[\begin{array}{cc} 2 & -1 \\ 1 & 1 \end{array}\right] \\ & A^2=\left[\begin{array}{cc} 3 & -3 \\ 3 & 0 \end{array}\right]=3\left[\begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array}\right] \\ & A^4=9\left[\begin{array}{ll} 0 & -1 \\ 1 & -1 \end{array}\right] \\ & A^8=81\left[\begin{array}{ll} -1 & 1 \\ -1 & 0 \end{array}\right] \\ & A^{12}=729\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & A^{13}=729\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 2 & -1 \\ 1 & 1 \end{array}\right] \\ & A^{13}=\left[\begin{array}{cc} 1458 & -729 \\ 729 & 729 \end{array}\right] \end{aligned}$$</p> <p>$$\begin{aligned} & \text { Sum }=2187=3^n \\ & 3^7=3^n \\ & n=7 \end{aligned}$$</p>