For two unimodular complex numbers $$z_1$$ and $$z_2$$, $$\begin{bmatrix} \overline z_1& -z_2 \\ \overline z_2 & z_1\end{bmatrix}^{-1}$$$$\begin{bmatrix} z_1& z_2 \\ -\overline z_2 & \overline z_1\end{bmatrix}^{-1}$$ is equal to
  • A
    $$\begin{bmatrix}

    z_1 & z_2\\

    \overline z_1 & \overline z_2

    \end{bmatrix}$$
  • A
    $$\begin{bmatrix}

    z_1 & z_2\\

    \overline z_1 & \overline z_2

    \end{bmatrix}$$
  • B
    $$\begin{bmatrix}

    1 & 0\\

    0 & 1

    \end{bmatrix}$$
  • B
    $$\begin{bmatrix}

    1 & 0\\

    0 & 1

    \end{bmatrix}$$
  • C
    $$\begin{bmatrix} \frac {1}{2}

    & 0\\ 0

    & \frac {1}{2}

    \end{bmatrix}$$
  • C
    $$\begin{bmatrix} \frac {1}{2}

    & 0\\ 0

    & \frac {1}{2}

    \end{bmatrix}$$
  • D
    $$\begin{bmatrix} 2

    & 0\\ 0

    & 2

    \end{bmatrix}$$
  • D
    $$\begin{bmatrix} 2

    & 0\\ 0

    & 2

    \end{bmatrix}$$

Solution

$$\begin{bmatrix} \overline z_1& -z_2 \\ \overline z_2 & z_1\end{bmatrix}^{-1}$$$$\begin{bmatrix} z_1& z_2 \\ -\overline z_2 & \overline z_1\end{bmatrix}^{-1}$$
$$\frac {1}{2} \frac {1}{2}$$$$\begin{bmatrix} z_1& z_2 \\ -\overline z_2 & \overline z_1\end{bmatrix}\begin{bmatrix} \overline z_1& -z_2 \\ \overline z_2 & z_1\end{bmatrix}=\frac {1}{4}\begin{bmatrix} 2 &0 \\ 0 & 2\end{bmatrix}=\begin{bmatrix} \frac {1}{2}&0 \\ 0 & \frac {1}{2}\end{bmatrix}$$