Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$

Question

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$, where $aj+b$ is a complex number, and $|f(x)|$ is the modulus function.

In the past I've been calculating $|(aj+b)^{-1}|$ by multiplying the numerator and denominator by the complex conjugate, so that I can get it in the form $|cj+d|$. But if the answer to this question is yes, then this will save me a lot of time.

Is there also a way to calculate the argument of $|(aj+b)^{-1}|$ without having to multiply by the complex conjugate and put in the form $|cj+d|$?

Answer 1

Yes:

$|\frac{z_1}{z_2}|=\frac{|z_1|}{|z_2|}\Rightarrow |\frac{1}{aj+b}|=\frac{|1|}{|aj+b|}=|aj+b|^{-1}$