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माना $A=\left(\begin{array}{cc}1+ i & 1 \\ - i & 0\end{array}\right)$ है जहाँ $i =\sqrt{-1}$ है।तब समुच्चय $\left\{ n \in\{1,2, \ldots, 100\}: A ^{ n }= A \right\}$ में अवयवों की संख्या है।
$255$
$25$
$75$
$80$
Solution
$A =\left[\begin{array}{cc}1+ i & 1 \\ – i & 0\end{array}\right]$
$A ^{2}=\left[\begin{array}{cc}1+ i & 1 \\ – i & 0\end{array}\right]\left[\begin{array}{cc}1+ i & 1 \\ – i & 0\end{array}\right]$
$A ^{2}=\left[\begin{array}{cc} i & 1+ i \\ – i +1 & – i \end{array}\right]$
$A ^{4}=\left[\begin{array}{cc} i & 1+ i \\ – i +1 & – i \end{array}\right]\left[\begin{array}{cc} i & 1+ i \\ – i +1 & – i \end{array}\right]$
$A ^{4}=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]= I$
$A ^{4 n +1}= A$
$n =1,5,9, \ldots \ldots, 97$
$\Rightarrow$ total elements in the set is $25 .$
