Question

Given that $i=\sqrt{-1}$, find the multiplicative inverse of $5-i$.

• A. $5+i$
• B. $\frac{5+i}{26}$
• C. $\frac{1}{5+i}$
• D. $\frac{5+i}{24}$
• E. $\frac{5-i}{24}$

Answer 1

The correct option is B $\frac{5+i}{26}$

Let the multiplicative inverse of $5-i$ be $a+bi$

Therefore we have $(5-i)(a+bi)=1$

$\Rightarrow 5a+b+i(5b-a)=1$

$\Rightarrow 5a+b=1$ and $5b-a=0$

$\Rightarrow a=5b$

Substitute this in $5a+b=1$, we get

$b=\frac{1}{26}$ and $a=\frac{5}{16}$

Therefore, multiplicative inverse is $\frac{(5+i)}{26}$.