Question

If $A=\left[\begin{array}{ccc}2& 3-i& -i\\ 3+i& -5& 7+i\\ i& 7-i& e\end{array}\right]$, then $A$ is

• A. symmetric
• B. Hermitian
• C. skew Hermitian
• D. none of these

Answer 1

The correct option is B Hermitian

$A=\left[\begin{array}{ccc}2& 3-i& -i\\ 3+i& -5& 7+i\\ i& 7-i& e\end{array}\right]\Rightarrow \overline{A}$

$=\left[\begin{array}{ccc}\overline{2}& \overline{3-i}& \overline{-i}\\ \overline{3+i}& \overline{-5}& \overline{7+i}\\ \overline{i}& \overline{7-i}& \overline{e}\end{array}\right]$

$=\left[\begin{array}{ccc}2& 3+i& i\\ 3-i& -5& 7-i\\ -i& 7+i& e\end{array}\right]\Rightarrow A^{\ast }$

$={\overline{A}}^T\left[\begin{array}{ccc}2& 3-i& -i\\ 3+i& -5& 7+i\\ i& 7-i& e\end{array}\right]$

Hence $A$ is hermitian