A) \[{{2}^{100}}A\]
B) \[{{2}^{99}}A\]
C) \[{{2}^{101}}A\]
D) None of these
Correct Answer: B
Solution :
\[A=\left[ \,\begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix}\, \right]\] \[{{A}^{2}}=\left[ \,\begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix}\, \right]\,\left[ \,\begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix}\, \right]\]= \[\left[ \,\begin{matrix} 2 & 2 \\ 2 & 2 \\ \end{matrix}\, \right]=2\left[ \,\begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix}\, \right]\] \[{{A}^{3}}=2\,\left[ \,\begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix}\, \right]\,\left[ \,\begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix}\, \right]={{2}^{2}}\left[ \,\begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix}\, \right]\] \[{{A}^{n}}={{2}^{n-1}}\left[ \,\begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix}\, \right]\] Þ \[{{A}^{100}}={{2}^{99}}\left[ \begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix} \right]\].
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