Question

The multiplicative inverse of $ z $ is:

• A. \( \frac{1}{z + \bar{z}} \)
• B. \( \frac{z}{|z|} \)
• C. \( \frac{\bar{z}}{|z|^2} \)
• D. \( \frac{1}{\bar{z}} \)

Hint:

To find the multiplicative inverse of a complex number \( z \), multiply numerator and denominator by its conjugate: \[ \frac{1}{z} = \frac{\bar{z}}{|z|^2} \]

Answer 1

The Correct Option is C

Step 1: Let \( z = a + ib \), where \( a, b \in \mathbb{R} \).
Then, the multiplicative inverse of \( z \), denoted by \( z^{-1} \), satisfies: \[ z \cdot z^{-1} = 1 \] Step 2: Multiply numerator and denominator by conjugate of \( z \): \[ \frac{1}{z} = \frac{1}{z} \cdot \frac{\bar{z}}{\bar{z}} = \frac{\bar{z}}{z\bar{z}} = \frac{\bar{z}}{|z|^2} \] Hence, the multiplicative inverse of \( z \) is: \[ \frac{\bar{z}}{|z|^2} \]