Question

If the matrix $A=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& i\\ -i& a\end{array}\right]$ is unitary, then $-a=$

Answer 1

Given , $A=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& i\\ -i& a\end{array}\right]$
$A^T=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& -i\\ i& a\end{array}\right]$

$A^{\ast }=\overline{\left(A^T\right)}=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& i\\ -i& a\end{array}\right]$

$A^{\ast }A=I$

$\frac{1}{2}\left[\begin{array}{cc}1& i\\ -i& a\end{array}\right]\left[\begin{array}{cc}1& i\\ -i& a\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\frac{1}{2}\left[\begin{array}{cc}2& i+ai\\ -i-ai& 1+a^2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\Rightarrow a+1=0$ or $a^2+1=2$

$\Rightarrow a=-1$