The complex conjugate $\overline A$ is obtained by conjugating every entry: $(\overline A)_{ij}=\overline{a_{ij}}$. For example, $\overline{\begin{bmatrix}1+i&2i\end{bmatrix}}=\begin{bmatrix}1-i&-2i\end{bmatrix}$. A real matrix satisfies $\overline A=A$. Conjugation does not exchange rows and columns; the adjoint does.
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