Question

Find the multiplicative inverse of $2-3i$.

Answer 1

Find the multiplicative inverse of complex number :

Let $z=2-3i$

The multiplicative inverse of a complex number is its reciprocal.

So, $\frac{1}{z}=\frac{1}{2-3i}$

$$\begin{gathered} \Rightarrow z^{-1}=\frac{1}{2-3i}\times \frac{2+3i}{2+3i} \\ \Rightarrow z^{-1}=\frac{2+3i}{(2-3i)(2+3i)} \\ \Rightarrow z^{-1}=\frac{2+3i}{2^2-{\left(3i\right)}^2} \\ \Rightarrow z^{-1}=\frac{2+3i}{4+9}\{i=\sqrt{-1}\} \\ \Rightarrow z^{-1}=\frac{2}{13}+\frac{3i}{13} \end{gathered}$$

Therefore, the multiplicative inverse of $2-3i$ will be $\frac{2}{13}+\frac{3i}{13}$.