Mathematics - Matrices Question with Solution | TestHub

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