Find the multiplicative inverse of the complex number –i

Let $z=-i$

Then, $\bar{z}=i$ and $|z|^{2}=1^{2}=1$

Therefore, the multiplicative inverse of $-i$ is given by

$z^{-1}=\frac{\bar{z}}{|z|^{2}}=\frac{i}{1}=i$