If A=pmatrix&i&pmatrix, =24, and A^5=pmatrixa&b&dpmatrix, where i=-1, then which one of th…

Question

If $A=\begin{pmatrix}\cos	heta&i\sin	heta\i\sin	heta&\cos	heta\end{pmatrix}$, $	heta=\dfrac\pi{24}$, and $A^5=\begin{pmatrix}a&b\c&d\end{pmatrix}$, where $i=\sqrt{-1}$, then which one of the following is NOT true?

Accepted Answer

By induction, $A^n=\begin{pmatrix}\cos n	heta & i\sin n	heta\i\sin n	heta&\cos n	heta\end{pmatrix}$, so $a=\cos5	heta,\ b=c=i\sin5	heta,\ d=\cos5	heta$.

$a^2-b^2=\cos^25	heta-(i\sin5	heta)^2=\cos^25	heta+\sin^25	heta=1$ --- always $1$, never $	frac12$. So (D) is false (not true), while (A),(B),(C) all check out ($a^2-c^2=1$ similarly, $a^2-d^2=0$ trivially, and $a^2+b^2=\cos10	heta\in[0,1]$ for the given $	heta$-range).

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