All the elements in a matrix A are complex numbers with imaginary parts not eq

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Question -

All the elements in a matrix A are complex numbers with imaginary parts not equal to zero. If $A^{*}$ is the conjugate of the matrix A, $a_{i j}$ is the general element of matrix A, then what is the general element of the matrix, $\frac{A + A^{*}}{2} .$

Options:

A .

2 l m (a_{i j})

B .

l m (a_{i j})

C .

R e (a_{i j})

D .

\frac{R e (a_{i j})}{2}

Answer: Option C
:
C

By taking complex conjugate of a matrix we reverse the sign of imaginary parts of all the elements in the original matrix. i.e., if the element in A is x + iy, then the corresponding element in $A^{*}$ is x - iy.
So when A and $A^{*}$ is added the imaginary parts cancel out and the sum becomes 2 times the real part of element in A.
i.e., since $(a_{i j})$ is general element in A, the general element in $A + A^{*}$ becomes $2 R e (a_{i j}) .$
$∴$ General element in $\frac{A + A^{*}}{2} = R e (a_{i j}) .$

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