**Conjugate of a matrix** is the matrix obtained from matrix ‘P’ on replacing its elements with the corresponding conjugate complex numbers. It is denoted by $\overline{P}$.

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## Conjugate of a matrix example

Let Q is a matrix such that

\[Q=\begin{bmatrix} 1+i &2+3i \\ 4-2i& 6 \end{bmatrix}\]

Now, to find the conjugate of this matrix Q, we find the conjugate of each element of matrix Q i.e.

\[\overline{Q}=\begin{bmatrix} 1-i &2-3i \\ 4+2i& 6 \end{bmatrix}\]

This is the conjugate of a 2 x 2 matrix Q.

## Conjugate of a matrix properties

The conjugate of matrices P and Q are

$\overline{P} \; and \; \overline{Q}, respectively$. Then,

$1. \; \overline{(\overline{P})}=P \; and \; \overline{(\overline{Q})}=Q$

$2. \; \overline{(P+Q)}=\overline{P}+\overline{Q}$

$3. \; \overline{(PQ)}=\overline{P} \; \overline{Q}$

$4. \; \overline{P}=-P$

If P is a purely imaginary matrix

$5. \; \overline{P}=P$

If P is a real matrix

$6. \; \overline{(aP)}=\overline{a} \; \overline{P}$

where ‘a’ is any complex number.