Mathematics - Matrices Question with Solution | TestHub

Let $A=\left[\begin{array}{ll}1& 1\\ 1& 1\end{array}\right]$ $$\begin{array}{l}{\mathrm{A}}^2=\left[\begin{array}{ll}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{ll}1& 1\\ 1& 1\end{array}\right]=2\left[\begin{array}{ll}1& 1\\ 1& 1\end{array}\right]=2\mathrm{A}\\ {\mathrm{A}}^3=2^2\left[\begin{array}{ll}1& 1\\ 1& 1\end{array}\right],{\mathrm{A}}^4=2^3\left[\begin{array}{ll}1& 1\\ 1& 1\end{array}\right]\\ {\mathrm{A}}^3=2^2\mathrm{A},{\mathrm{A}}^4=2^3\mathrm{A}\\ \therefore {\mathrm{A}}^{\mathrm{n}}=2^{\mathrm{n}-1}\left[\begin{array}{ll}1& 1\\ 1& 1\end{array}\right]\\ \Rightarrow {\mathrm{A}}^{100}=2^{100-1}\mathrm{A}\therefore {\mathrm{A}}^{100}=2^{99}\mathrm{A}\end{array}$$