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If $A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$ then $A^{100}=$
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$2^{99} \mathrm{~A}$
$A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
$A^2=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$$=\left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]=$ 2 $\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
$A^3=2\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$ $\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$ $=2^2\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
$A^n=2^{n-1}\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$ $\Rightarrow A^{100}=2^{99}\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
$A^2=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$$=\left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]=$ 2 $\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
$A^3=2\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$ $\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$ $=2^2\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
$A^n=2^{n-1}\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$ $\Rightarrow A^{100}=2^{99}\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
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