设 $\mathrm{F}_4=\left[\begin{array}{cccc}1 & 1 & 1 & 1 \\ 1 & i & -1 & -i \\ 1 & -1 & 1 & -1 \\ 1 & -i & -1 & i\end{array}\right], \mathrm{F}_2=\left[\begin{array}{cc}1 & 1 \\ 1 & -1\end{array}\right], \mathrm{D}_2=\left[\begin{array}{ll}1 & 0 \\ 0 & i\end{array}\right]$ ．
1）求矩阵 $\mathbf{C}$ ，使得 $\left[\begin{array}{cc}\mathbf{I}_2 & \mathbf{D}_2 \\ \mathbf{I}_2 & -\mathbf{D}_2\end{array}\right]\left[\begin{array}{cc}\mathbf{F}_2 & \mathbf{0} \\ \mathbf{0} & \mathbf{F}_2\end{array}\right] \mathbf{C}=\mathrm{F}_4$ ；
2）求 $\mathrm{F}_4$ 的逆矩阵．