A) \[\left[ \begin{matrix} {{z}_{1}} & {{z}_{2}} \\ {{{\bar{z}}}_{1}} & {{{\bar{z}}}_{2}} \\ \end{matrix} \right]\]
B) \[\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]\]
C) \[\left[ \begin{matrix} 1/2 & 0 \\ 0 & 1/2 \\ \end{matrix} \right]\]
D) None of these
Correct Answer: C
Solution :
\[{{\left[ \begin{matrix} {{{\bar{z}}}_{1}} & -{{z}_{2}} \\ {{{\bar{z}}}_{2}} & {{z}_{1}} \\ \end{matrix} \right]}^{-1}}{{\left[ \begin{matrix} {{z}_{1}} & {{z}_{2}} \\ -{{{\bar{z}}}_{2}} & {{{\bar{z}}}_{1}} \\ \end{matrix} \right]}^{-1}}\] \[={{\left\{ \left[ \begin{matrix} {{z}_{1}} & {{z}_{2}} \\ -{{{\bar{z}}}_{2}} & {{{\bar{z}}}_{1}} \\ \end{matrix} \right]\left[ \begin{matrix} {{{\bar{z}}}_{1}} & -{{z}_{2}} \\ {{{\bar{z}}}_{2}} & {{z}_{1}} \\ \end{matrix} \right] \right\}}^{-1}}\] \[={{\left[ \begin{matrix} {{z}_{1}}{{{\bar{z}}}_{1}}+{{z}_{2}}{{{\bar{z}}}_{2}} & 0 \\ 0 & {{z}_{2}}{{{\bar{z}}}_{2}}+{{z}_{1}}{{{\bar{z}}}_{1}} \\ \end{matrix} \right]}^{-1}}\] \[={{\left[ \begin{matrix} |{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}} & 0 \\ 0 & |{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}} \\ \end{matrix} \right]}^{-1}}\] \[={{\left[ \begin{matrix} 2 & 0 \\ 0 & 2 \\ \end{matrix} \right]}^{-1}}=\left[ \begin{matrix} 1/2 & 0 \\ 0 & 1/2 \\ \end{matrix} \right]\]
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