Question

Which among the following is/are skew hermitian matrix.

• A. $\left[\begin{array}{cc}o& i\\ i& 0\end{array}\right]$
• B. $\left[\begin{array}{ccc}i& 3+i& 5-i\\ -3+i& -i& 0\\ -5-i& 0& 3i\end{array}\right]$
• C. $\left[\begin{array}{cc}i+1& 3i\\ 3i& i+1\end{array}\right]$
• D. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Answer 1

Answer: (A), (B)

$\left[\begin{array}{ccc}i& 3+i& 5-i\\ -3+i& -i& 0\\ -5-i& 0& 3i\end{array}\right]$

We already saw that a square matrix is skew hermitian if its conjugate transpose is equal to its negative.

Lets look at the matrices given. Note that (a) and (b) satisfy this condition. For (c) both diagonal elements are 1+i. If we take conjugate and take its negative it becomes –(1 – i) or i – 1, which is not equal to 1+i. So it’s not skew hermitian.

Similarly in option (d) diagonal elements are 1. If you take its conjugate and negative you get -1. So an identity matrix cannot be skew hermitian.

So the correct options are (a) and (b).