20.
設\( \displaystyle Z=\Bigg[\; \matrix{cos120^o & -sin120^o \cr sin120^o & cos120^o} \Bigg]\; \)，則\( Z+Z^2+Z^3+...+Z^{100}= \)
①\( \displaystyle \Bigg[\; \matrix{1 & 0 \cr 0 & 1} \Bigg]\; \) ②\( \displaystyle \Bigg[\; \matrix{0 & 0 \cr 0 & 0} \Bigg]\; \) ③\( \displaystyle \Bigg[\; \matrix{\frac{-1}{2} & \frac{-\sqrt{3}}{2} \cr \frac{\sqrt{3}}{2} & \frac{-1}{2}} \Bigg]\; \) ④\( \displaystyle \Bigg[\; \matrix{\frac{-1}{2} & \frac{\sqrt{3}}{2} \cr \frac{-\sqrt{3}}{2} & \frac{-1}{2}} \Bigg]\; \)
[解答]
這題是考旋轉矩陣
\( \displaystyle Z=\Bigg[\; \matrix{cos120^o & -sin120^o \cr sin120^o & cos120^o} \Bigg]\; \)
\( \displaystyle Z^2=\Bigg[\; \matrix{cos240^o & -sin240^o \cr sin240^o & cos240^o} \Bigg]\; \)
\( \displaystyle Z^3=\Bigg[\; \matrix{cos360^o & -sin360^o \cr sin360^o & cos360^o} \Bigg]\;=I_2 \)
\( Z+Z^2+Z^3=\Bigg[\; \matrix{0 & 0 \cr 0 & 0} \Bigg]\; \)
\( (Z+Z^2+Z^3)+(Z^4+Z^5+Z^6)+...+(Z^{97}+Z^{98}+Z^{99})+Z^{100}=Z^1 \)
答案③
[ 本帖最後由 bugmens 於 2011-9-19 07:55 AM 編輯 ]
