证明: 存在 $n$ 阶实方阵 $\boldsymbol{A}$, 使得
$$
\sin \boldsymbol{A}=\left(\begin{array}{ccccc}
\frac{1}{2} & \frac{1}{4} & \cdots & \cdots & \frac{1}{2^n} \\
& \frac{1}{2} & \frac{1}{4} & \cdots & \frac{1}{2^{n-1}} \\
& & \ddots & \ddots & \vdots \\
& & & \ddots & \frac{1}{4} \\
& & & & \frac{1}{2}
\end{array}\right) .
$$