Question

The multiplicative inverse of $ \frac{3 + 4i}{4 - 5 i}$ is

• A. $\left(\frac{-8}{25} , \frac{31}{25} \right)$
• B. $\left(\frac{-8}{25} , \frac{-31}{25} \right)$
• C. $\left(\frac{8}{25} , \frac{-31}{25} \right)$
• D. $\left(\frac{8}{25} , \frac{31}{25} \right)$

Answer 1

Answer: (B)

Let $z=\frac{3+4i}{4-5i}$
we have to calculate $z^{-1} \, i.e., \frac{1}{z}$
$ \therefore \:\:\: z^{-1} =\frac{1}{z} = \frac{4-5i}{3+4i}\times\frac{3-4i}{3-4i}$
$ = \frac{12-15i -16i+20i^{2}}{9-16i^{2}} =\frac{12-31i-20}{9-16\left(-1\right)} $
$= \frac{-8-31i}{9+16}=\frac{-8}{25} -\frac{31}{25}i$
$\therefore \:\:\:\: z^{-1}=\left( \frac{-8}{25} , \frac{-31}{25}\right)$