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

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 . 2lm(aij)

B . lm(aij)

C . Re(aij)

D . Re(aij)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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