Conjugate of a Matrix - Practice Questions & MCQ

If a matrix A has complex numbers as its elements, then the matrix obtained by replacing those complex numbers by their conjugates is called the conjugate of the matrix A and it is denoted by $\overline{\mathrm{A}}$. (If the element of a matrix is a + ib, then it is replaced by a - ib .)

$\begin{aligned} \text { e.g. } \mathrm{A} & =\left[\begin{array}{ccc}2 i & 3+4 i & 7 \\ 3 i & 9 & 4+5 i \\ 4+5 i & 4 i & 3+7 i\end{array}\right] \text { then, } \\ \overline{\mathrm{A}} & =\left[\begin{array}{ccc}-2 i & 3-4 i & 7 \\ -3 i & 9 & 4-5 i \\ 4-5 i & -4 i & 3-7 i\end{array}\right]\end{aligned}$

Properties of the conjugate of a matrix:

If A and B are two matrices of the same order, then

i) $\overline{(\overline{\mathrm{A}})}=\mathrm{A}$
ii) $\overline{(\mathrm{A}+\mathrm{B})}=\overline{\mathrm{A}}+\overline{\mathrm{B}}_{\text {where } \mathrm{A} \text { and } \mathrm{B} \text { are conformable for matrix addition. }}$
iii) $\overline{(\mathrm{A} \times \mathrm{B})}=\overline{\mathrm{A}} \times \overline{\mathrm{B}}$ where A and B are conformable for multiplication.
$\mathrm{iv)} \overline{(\mathrm{kA})}=\overline{\mathrm{k}} \cdot \overline{\mathrm{A}}_{\text {, where } \mathrm{k} \text { is real or complex. }}$

The transpose conjugate of a matrix and properties:

The transpose of a conjugate matrix A is called the transposed conjugate of A and is denoted by A^?. The conjugate of the transpose of A is the same as the transpose of the conjugate of A

$\begin{aligned} & \text { i.e. } A^\theta=(\overline{\mathrm{A}})^{\prime}=\overline{\left(\mathrm{A}^{\prime}\right)} \\ & \mathrm{A}=\left[\begin{array}{ccc}1+2 i & 3 i & 5+4 i \\ 2 i-1 & 1-i & 0 \\ 3+i & 1+i & 12\end{array}\right] \\ & \overline{\mathrm{A}}=\left[\begin{array}{ccc}1-2 i & -3 i & 5-4 i \\ -2 i-1 & 1+i & 0 \\ 3-i & 1-i & 12\end{array}\right] \\ & (\overline{\mathrm{A}})^{\prime}=\left[\begin{array}{ccc}1-2 i & -2 i-1 & 3-i \\ -3 i & 1+i & 1-i \\ 5-4 i & 0 & 12\end{array}\right]\end{aligned}$

Properties of the transpose conjugate matrix:

If A and B are two matrices of the same order then

i) conjugate of a conjugate of matrix is the same as the original matrix itself,

In mathematical language (A^?)^? = A, which is quite obvious as we are reversing back the things which we did while taking conjugate at first time.

ii) (A + B)^? = A^?  + B^?, this is obvious if a matrix is conformable, as addition is done element-wise.

Iii )(kA)^? = kA^?, since multiplication in a matrix are elementwise, hence this is also obvious, as all elements multiplied before conjugate will be amplified in the way as after taking conjugate.

iv) (AB)^? = B^?A^?, here A and B should be conformable for matrix multiplication.