For each of the following matrices, find its transpose and state whether it is symmetric, skew- symmetric or neither. [01+2ii-2-1-2i0-72-i70]

Exercise 2.2 | Q 9.3 | Page 47

QUESTION

For each of the following matrices, find its transpose and state whether it is symmetric, skew- symmetric or neither.

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

${\displaystyle }$

SOLUTION

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

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

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

^∴ A^T = – A i.e. A = – A^T
∴ A is a skew-symmetric matrix.