Question

If $A=\left[\begin{array}{ccc}1& 2-3i& 3+4i\\ 2+3i& 0& 4-5i\\ 3-4i& 4+5i& 2\end{array}\right]$ then A is

• A. Hermitian
• B. Skew-Hermitian
• C. Symmetric
• D. Skew-Symmetric

Answer 1

Answer: (A)

Hermitian

$A=\left[\begin{array}{ccc}1& 2-3i& 3+4i\\ 2+3i& 0& 4-5i\\ 3-4i& 4+5i& 2\end{array}\right]$
$\overline{A}=\left[\begin{array}{ccc}1& 2+3i& 3-4i\\ 2-3i& 0& 4+5i\\ 3+4i& 4-5i& 2\end{array}\right]$
$\Rightarrow ({\overline{A})}^T=\left[\begin{array}{ccc}1& 2-3i& 3+4i\\ 2+3i& 0& 4-5i\\ 3-4i& 4+5i& 2\end{array}\right]$
So, $({\overline{A})}^T=A$
Hence, A is Hermitian