Question

Consider the following in respect of a complex number z:

1. \(\rm {\overline{\left(z^{-1}\right)}}=(\bar{z})^{-1}\)

2. zz^-1 = |z|²

Which of the above is/are correct?

1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2

Answer 1

Correct Answer - Option 1 : 1 only

Concept:

Consider z = a + ib

Multiplicative identity z.z^-1 = 1

zz̅ = a² + b² = (a + ib) (a - ib)

 

Explanation

Given:

1. \(\rm {\overline{\left(z^{-1}\right)}}=(\bar{z})^{-1}\)

consider z = a - ib

⇒ z^-1 = \(\rm \frac 1 z = \frac 1 {a - ib}\) =\(\rm \frac{a + ib}{a^{2} + b^{2}}\)

⇒ \(\rm \overline{z^{-1}}\) = \(\rm \frac{a - ib}{a^{2} + b^{2}}\)      ....(1)

⇒ \(\rm \overline{z}\) = a + ib

⇒ (\(\rm \overline{z}\))^-1 = \(\rm \frac 1 {\bar{z}} = \frac 1 {a +ib}\)= \(\rm \frac{a - ib}{a^{2} + b^{2}}\)      ....(2)

from eq(1) and (2)

statement 1 is correct. 

 

2. zz-1 = |z|2

consider z = a - ib

⇒ z^-1 = \(\rm \frac{a + ib}{a^{2} + b^{2}}\)

⇒ z.z-1 = (a - ib)\(\rm \frac{a + ib}{a^{2} + b^{2}}\) = \(\rm \frac{a^{2} + b^{2}}{a^{2} + b^{2}} =1\)

Statement 2 is not correct