Question

Let $A=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right],i=\sqrt{-1}$. Then, the system of linear equations $A^8\left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{c}8\\ 64\end{array}\right]$ has :

• A. A unique solution
• B. Infinitely many solutions
• C. No solution
• D. Exactly two solutions

Answer 1

Answer: (C)

$A=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right]$
$A^2=\left[\begin{array}{cc}-2& 2\\ 2& -2\end{array}\right]=2\left[\begin{array}{cc}-1& 1\\ 1& -1\end{array}\right]$
$A^4=2^2\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]=8\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$
$A^8=64\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]=128\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$
$A^8\left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{c}8\\ 64\end{array}\right]$
$\Rightarrow 128\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{c}8\\ 64\end{array}\right]$
$\Rightarrow 128\left[\begin{array}{c}x-y\\ -x+y\end{array}\right]=\left[\begin{array}{c}8\\ 64\end{array}\right]$
$\Rightarrow x-y=\frac{1}{16}\dots ..(1)$
$\&-x+y=\frac{1}{2}\dots ..(2)$
$\Rightarrow$ From (1) & (2): No solution.