Question

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

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

Answer 1

The correct option is D $\left(z\overline{z}\right)$ is nonnegative real

Let $z=x+iy\Rightarrow \overline{z}=x-iy$

$z-\overline{z}=2iy$

$z+\overline{z}=2x$

$z\overline{z}=x^2+y^2$
Hence $z\overline{z}$ is a nonnegative real