Q. For two unimodular complex numbers $\mathcal{- z}_{2}$ and$\mathcal{z}_{2}$, $\begin{bmatrix} \overline{\mathcal{z}_{1}} & \mathcal{- z}_{2} \\ \overline{\mathcal{z}_{2}} & \mathcal{z}_{1} \end{bmatrix}^{- 1}\begin{bmatrix} \mathcal{z}_{1} & \mathcal{z}_{2} \\ - \ \overline{\mathcal{z}_{2}} & \overline{\mathcal{z}_{1}} \end{bmatrix}^{- 1}$ is equal to
Explanation
(c)
$\begin{bmatrix}
{\overline{\mathcal{z}}}_{1} & \mathcal{- z}_{2} \\
{\overline{\mathcal{z}}}_{1} & \mathcal{z}_{1}
\end{bmatrix}^{- 1}\begin{bmatrix}
\mathcal{z}_{1} & \mathcal{z}_{2} \\
- {\overline{\mathcal{z}}}_{2} & \mathcal{z}_{1}
\end{bmatrix}^{- 1}$
$= \left( \begin{bmatrix}
\mathcal{z}_{1} & \mathcal{z}_{2} \\
- {\overline{\mathcal{z}}}_{2} & {\overline{\mathcal{z}}}_{1}
\end{bmatrix}\begin{bmatrix}
\overline{z_{1}} & - z_{2} \\
{\overline{\mathcal{z}}}_{2} & \mathcal{z}_{1}
\end{bmatrix} \right)^{- 1}$
$= \begin{bmatrix}
\mathcal{z}_{1}{\overline{\mathcal{z}}}_{1} & 0 \\
0 & \mathcal{z}_{2}{\overline{\mathcal{z}}}_{2} + \mathcal{z}_{1}{\overline{\mathcal{z}}}_{1}
\end{bmatrix}^{- 1}$
$= \begin{bmatrix}
\left| \mathcal{z}_{1} \right|^{2}{+ \left| \mathcal{z}_{2} \right|}^{2} & 0 \\
0 & \left| \mathcal{z}_{1} \right|^{2}{+ \left| \mathcal{z}_{2} \right|}^{2}
\end{bmatrix}^{- 1}$
$= \begin{bmatrix}
2 & 0 \\
0 & 2
\end{bmatrix}^{- 1} = \begin{bmatrix}
\frac{1}{2} & 0 \\
0 & \frac{1}{2}
\end{bmatrix}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.