Question

For two unimodular complex numbers $z_1$ and $z_2$, ${\left[\begin{array}{cc}\overline{z_1}& -z_2\\ \overline{z_2}& z_1\end{array}\right]}^{-1}{\left[\begin{array}{cc}{z}_1& z_2\\ -\overline{z_2}& \overline{z_1}\end{array}\right]}^{-1}$ is equal to

• A. $\left[\begin{array}{cc}{z}_1& z_2\\ \overline{z_1}& \overline{z_2}\end{array}\right]$
• B. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
• C. $\left[\begin{array}{cc}1/2& 0\\ 0& 1/2\end{array}\right]$
• D. none of these

Answer 1

The correct option is C $\left[\begin{array}{cc}1/2& 0\\ 0& 1/2\end{array}\right]$

Given, $|z_1|=|z_2|=1$
$z_1$ and $z_2$, ${\left[\begin{array}{cc}\overline{z_1}& -z_2\\ \overline{z_2}& z_1\end{array}\right]}^{-1}{\left[\begin{array}{cc}{z}_1& z_2\\ -\overline{z_2}& \overline{z_1}\end{array}\right]}^{-1}$
Let $A=\left[\begin{array}{cc}\overline{z_1}& -z_2\\ \overline{z_2}& z_1\end{array}\right],B=\left[\begin{array}{cc}{z}_1& z_2\\ -\overline{z_2}& \overline{z_1}\end{array}\right]$
Here, $|A|=z_1\overline{z_1}+z_2\overline{z_2}=|z_1{|}^2+|z_2{|}^2=2$
$|B|=z_1\overline{z_1}+z_2\overline{z_2}=|z_1{|}^2+|z_2{|}^2=2$
Hence, $A^{-1},B^{-1}$ exists
Now, $adjA=C^T={\left[\begin{array}{cc}{z}_1& -\overline{z_2}\\ z_2& \overline{z_1}\end{array}\right]}^T$
$adjA=\left[\begin{array}{cc}{z}_1& z_2\\ -\overline{z_2}& \overline{z_1}\end{array}\right]$
Hence, $A^{-1}=\frac{1}{2}\left[\begin{array}{cc}{z}_1& z_2\\ -\overline{z_2}& \overline{z_1}\end{array}\right]$
Now, $adjB={\left[\begin{array}{cc}\overline{z_1}& \overline{z_2}\\ z_2& z_1\end{array}\right]}^T$
$adjB=\left[\begin{array}{cc}\overline{z_1}& z_2\\ \overline{z_2}& z_1\end{array}\right]$
Hence,$B^{-1}=\frac{1}{2}\left[\begin{array}{cc}\overline{z_1}& z_2\\ \overline{z_2}& z_1\end{array}\right]$
Now, $A^{-1}{B}^{-1}=\frac{1}{4}\left[\begin{array}{cc}{z}_1& z_2\\ -\overline{z_2}& \overline{z_1}\end{array}\right]\left[\begin{array}{cc}\overline{z_1}& z_2\\ \overline{z_2}& z_1\end{array}\right]$
$=\frac{1}{4}\left[\begin{array}{cc}|z_1{|}^2+|z_2{|}^2& 0\\ 0& |z_1{|}^2+|z_2{|}^2\end{array}\right]$
$=\left[\begin{array}{cc}\frac{1}{2}& 0\\ 0& \frac{1}{2}\end{array}\right]$