Question

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

• A. $2lm\left(a_{ij}\right)$
• B. $lm\left(a_{ij}\right)$
• C. $Re\left(a_{ij}\right)$
• D. $\frac{Re\left(a_{ij}\right)}{2}$

Answer 1

The correct option is C $Re\left(a_{ij}\right)$

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 $A^{\ast }$ is x - iy.
So when A and $A^{\ast }$ is added the imaginary parts cancel out and the sum becomes 2 times the real part of element in A.
i.e., since $\left(a_{ij}\right)$ is general element in A, the general element in $A+A^{\ast }$ becomes $2Re\left(a_{ij}\right).$
$\therefore$ General element in $\frac{A+A^{\ast }}{2}=Re\left(a_{ij}\right).$