行列$\displaystyle A=\frac{1}{2} \left( \begin{array}{cc} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{array} \right)$を考える．また，$E$を単位行列とする．
(1) $A=\left( \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right) \ \ (0 \leqq \theta<2\pi)$と表すと，$\displaystyle \theta=\frac{\fbox{ア}}{\fbox{イ}}$である．
(2) $E+A+A^2=\left( \begin{array}{cc} \fbox{ウ} & -\sqrt{\fbox{エ}} \\ \sqrt{\fbox{オ}} & \fbox{カ} \end{array} \right)$，$A^3=\left( \begin{array}{cc} \fbox{キ}\fbox{ク} & \fbox{ケ} \\ \fbox{コ} & \fbox{サ}\fbox{シ} \end{array} \right)$，$E+A+A^2+A^3+A^4+A^5=\left( \begin{array}{cc} \fbox{ス} & \fbox{セ} \\ \fbox{ソ} & \fbox{タ} \end{array} \right)$である．
(3) $E+A+A^2+A^3+\cdots +A^{20}=\left( \begin{array}{cc} \fbox{チ} & -\sqrt{\fbox{ツ}} \\ \sqrt{\fbox{テ}} & \fbox{ト} \end{array} \right)$である．