Question

If $A=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1& i+1\\ i-1& 1\end{array}\right]$, then $A\left(\stackrel{\_}{A^T}\right)$equals

• A. $0$
• B. $I$
• C. $-I$
• D. $2I$

Answer 1

Answer: (B)

$I$

$A=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1& i+1\\ i-1& 1\end{array}\right]$
$A^T=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1& i-1\\ i+1& 1\end{array}\right]$
$\overline{A^T}=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1& -i-1\\ 1-i& 1\end{array}\right]$
Now, $A\overline{A^T}=\frac{1}{3}\left[\begin{array}{cc}1& i+1\\ i-1& 1\end{array}\right]\left[\begin{array}{cc}1& -i-1\\ 1-i& 1\end{array}\right]$
$=\frac{1}{3}\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$
$=I$
$\Rightarrow A\overline{A^T}=I$