Q. Let $A=\begin{pmatrix}1+ i & 1 \\ - i & 0\end{pmatrix}$ where $i =\sqrt{-1}$. Then, the number of elements in the set $\left\{ n \in\{1,2, \ldots, 100\}: A ^{ n }= A \right\}$ is

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A

B

C

D

Solution:

$A=\begin{bmatrix}1+ i & 1 \\ - i & 0\end{bmatrix}$
$A ^{2}=\begin{bmatrix}1+ i & 1 \\ - i & 0\end{bmatrix}\begin{bmatrix}1+ i & 1 \\ - i & 0\end{bmatrix}$
$A ^{2}=\begin{bmatrix}i & 1+ i \\ - i +1 & - i \end{bmatrix}$
$A ^{4}=\begin{bmatrix}i & 1+ i \\ - i +1 & - i \end{bmatrix}\begin{bmatrix} i & 1+ i \\ - i +1 & - i \end{bmatrix}$
$A ^{4}=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}= I$
$A ^{4 n +1}= A$
$n =1,5,9, \ldots \ldots, 97$
$\Rightarrow$ total elements in the set is $25 .$

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