Question

Find the multiplicative inverse of the complex number –i?

Answer 1

Hint: Here, we will find the solution using the formulae of multiplicative inverse of a complex number i.e. ${z^{ - 1}} = \dfrac{{\bar z }}{{{{\left| z \right|}^2}}}$.

Complete step-by-step answer:
We need to find multiplicative inverse of –i
So let $z = - i$
Now multiplicative inverse is ${z^{ - 1}}$, so ${z^{ - 1}} = \dfrac{{\bar z }}{{{{\left| z \right|}^2}}} \to (1)$
Conjugate of z is,
$\bar z = i$
Modulus of z is,
$\Rightarrow$ $\left| z \right| = \sqrt {\operatorname{Re} {{(z)}^2} + \operatorname{Im} g{{(z)}^2}} $
$\Rightarrow$ $\left| z \right| = \sqrt {{{(0)}^2} + {{( - 1)}^2}} = 1$
$\Rightarrow$ ${\left| z \right|^2} = 1$

Therefore
${z^{ - 1}} = \dfrac{i}{1} = i$
Hence, the answer to this problem is i.

Note: Whenever we are being asked to find the multiplicative inverse simply find ${z^{ - 1}}$ and use the formulae for ${z^{ - 1}}$ as mentioned above this will give you an answer.