Write the multiplicative inverse of complex number (-2+3 i). - Vidyadip

Question

Write the multiplicative inverse of complex number $(-2+3 i)$.

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Answer

Given, $z=-2+3 i$
$\begin{aligned}\frac{1}{z} & =z^{-1}=\frac{1}{-2+3 i} \\\therefore \quad z^{-1} & =\frac{1}{(-2+3 i)} \times \frac{(-2-3 i)}{(-2-3 i)}\end{aligned}$
[Multiplying numerator and denominator by ( $-2-3 i$ ]
$\begin{aligned}z^{-1} & =\frac{-2-3 i}{(-2)^2-(3 i)^2} \\& =\frac{-2-3 i}{4+9} \\z^{-1} & =\frac{-2}{13}-\frac{3}{13} i\end{aligned}$

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