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જો $A=\left[\begin{array}{cc}i & -i \\ -i & i\end{array}\right], i=\sqrt{-1}$ હોય તો સુરેખ સંહતિ સમીકરણો $A^{8}\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]$ એ . . . ઉકેલ ધરાવે. ..
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Solution
$A=\left[\begin{array}{cc}i & -i \\ -i & i\end{array}\right]$
$A^{2}=\left[\begin{array}{cc}-2 & 2 \\ 2 & -2\end{array}\right]=2\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right]$
$A^{4}=2^{2}\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]=8\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]$
$A^{8}=64\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]=128\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]$
$A^{8}\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}8 \\ 64\end{array}\right]$
$\Rightarrow 128\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]$
$\Rightarrow \quad 128\left[\begin{array}{c}x-y \\ -x+y\end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]$
$\Rightarrow \quad x-y=\frac{1}{16}………(1)$
$\quad-x+y=\frac{1}{2}$ $…..(2)$
$\Rightarrow$ From $(1)$ and $(2)$ No solution.
