Question

Find the transpose of $A$ and state whether it is a symmetric, a skew-symmetric or neither. $A=\left[\begin{array}{ccc}0& 1+2i& i-2\\ -1-2i& 0& -7\\ 2-i& 7& 0\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{ccc}0& 1+2i& i-2\\ -1-2i& 0& -7\\ 2-i& 7& 0\end{array}\right]$

$A^T=\left[\begin{array}{ccc}0& -1-2i& 2-i\\ 1+2i& 0& 7\\ i-2& -7& 0\end{array}\right]$

$=-\left[\begin{array}{ccc}0& 1+2i& i-2\\ -1-2i& 0& -7\\ 2-i& 7& 0\end{array}\right]=-A$

Since $A^T=-A$, the given matrix is a skew-symmetric matrix.