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

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

Aicharya · Accepted Answer

$A=\left[\begin{array}{cc}2 & -1 \ 1 & 1\end{array}ight]$ $\begin{aligned} & A^2=\left[\begin{array}{cc}2 & -1 \ 1 & 1\end{array}ight]\left[\begin{array}{cc}2 & -1 \ 1 & 1\end{array}ight]=\left[\begin{array}{cc}3 & -3 \ 3 & 0\end{array}ight] \ & A^3=\left[\begin{array}{cc}3 & -3 \ 3 & 0\end{array}ight]\left[\begin{array}{cc}2 & -1 \ 1 & 1\end{array}ight]=\left[\begin{array}{cc}3 & -6 \ 6 & -3\end{array}ight] \ & A^4=\left[\begin{array}{cc}3 & -6 \ 6 & -3\end{array}ight]\left[\begin{array}{cc}2 & -1 \ 1 & 1\end{array}ight]=\left[\begin{array}{cc}0 & -9 \ 9 & -9\end{array}ight] \ & A^5=\left[\begin{array}{cc}0 & -9 \ 9 & -9\end{array}ight]\left[\begin{array}{cc}2 & -1 \ 1 & 1\end{array}ight]=\left[\begin{array}{cc}-9 & -9 \ 9 & -18\end{array}ight] \ & A^6=\left[\begin{array}{cc}-9 & -9 \ 9 & -18\end{array}ight]\left[\begin{array}{cc}2 & -1 \ 1 & 1\end{array}ight]=\left[\begin{array}{cc}-27 & 0 \ 0 & -27\end{array}ight] \ & A^7=\left[\begin{array}{cc}-27 & -0 \ 0 & -27\end{array}ight]\left[\begin{array}{cc}-54 & 27 \ -27 & -27\end{array}ight]=\left[\begin{array}{cc}3^6 	imes 2 & -27^2 \ 27^2 & 3^6\end{array}ight] \ & 3^7=3^n \Rightarrow n=7\end{aligned}$

Let $A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$. If the sum of the diagonal elements of…

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