Let A=pmatrixi&-i-i&ipmatrix, i=-1.

Question

Let $A=\begin{pmatrix}i&-i\-i&i\end{pmatrix}$, $i=\sqrt{-1}$. Then the system of linear equations $A^8\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}8\64\end{pmatrix}$ has

Accepted Answer

$A^2=\begin{pmatrix}-2&2\2&-2\end{pmatrix}$, $A^4=\begin{pmatrix}8&-8\-8&8\end{pmatrix}$, $A^8=\begin{pmatrix}128&-128\-128&128\end{pmatrix}$.

The system becomes $128x-128y=8\Rightarrow x-y=\dfrac1{16}$, and $-128x+128y=64\Rightarrow x-y=-\dfrac12$.

These two conditions on $x-y$ are contradictory (parallel, inconsistent lines), so the system has no solution.

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