Mathematics - Matrices Question with Solution | TestHub

$A=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right]$

$A^2=\left[\begin{array}{cc}-2& 2\\ 2& -2\end{array}\right]=2\left[\begin{array}{cc}-1& 1\\ 1& -1\end{array}\right]$

$A^4=2^2\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]=8\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$

$A^8=64\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]=128\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$

$A^8\left[\begin{array}{l}\mathrm{x}\\ \mathrm{y}\end{array}\right]=\left[\begin{array}{c}8\\ 64\end{array}\right]$

$\Rightarrow 128\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{c}8\\ 64\end{array}\right]$

$\Rightarrow 128\left[\begin{array}{c}x-y\\ -x+y\end{array}\right]=\left[\begin{array}{c}8\\ 64\end{array}\right]$

$\Rightarrow x-y=\frac{1}{16}$   $...\left(1\right)$

and $-x+y=\frac{1}{2}$   $...\left(2\right)$

$\Rightarrow$ From $\left(1\right)$ and $\left(2\right)$ : No solution.