Question

If $z$ is a complex number, then which of the following statement is true?

• A. $\left(z\overline{z}\right)$ is purely imaginary
• B. $\left(z\overline{z}\right)$ is non-negative real
• C. $(z-\overline{z})$ is purely real
• D. $(z+\overline{z})$ is purely imaginary

Answer 1

The correct option is B $\left(z\overline{z}\right)$ is non-negative real

Let $z=x+iy$, then $\overline{z}=x-iy$
Now,
(i) $z\overline{z}=x^2+y^2$ [purely real]
(ii) $z\overline{z}=x^2+y^2$ [non-negative real]
(iii) $z-\overline{z}=2iIm\left(z\right)$ [purely imaginary]
(iv) $z+\overline{z}=2Re\left(z\right)$ [purely real]