Question

What is the transpose conjugate for the matrix. $\left[\begin{array}{cc}2+3i& -i\\ 5& 5-i\end{array}\right]$

• A. $\left[\begin{array}{cc}2-3i& -5\\ -i& 5+i\end{array}\right]$
• B. $\left[\begin{array}{cc}-2-3i& -5\\ i& -5+i\end{array}\right]$
• C. $\left[\begin{array}{cc}5-i& -i\\ 5& 2+3i\end{array}\right]$
• D. $\left[\begin{array}{cc}2-3i& 5\\ i& 5+i\end{array}\right]$

Answer 1

The correct option is D

$\left[\begin{array}{cc}2-3i& 5\\ i& 5+i\end{array}\right]$

Given matrix is $A=\left[\begin{array}{cc}2+3i& -i\\ 5& 5-i\end{array}\right]$

Taking conjugate transpose of matrix is equivalent to taking complex conjugate of each and every element of the matrix and transposing the resulting matrix. The order in which these are done doesn’t matter. In other words,

${\overline{\left[A\right]}}^T=\overline{\left[A^T\right]}$

Here taking conjugate first then the transpose

$\overline{A}=\left[\begin{array}{cc}2-3i& i\\ 5& 5+i\end{array}\right]{\overline{\left[A\right]}}^T=\left[\begin{array}{cc}2-3i& 5\\ i& 5+i\end{array}\right]=A^{\ast }$
Hence option (d) is correct.