Question

If $z=1+i$, then the multiplicative inverse of $z^2$ is

• A. $2i$
• B. $1-i$
• C. $-\frac{i}{2}$
• D. $\frac{i}{2}$

Answer 1

The correct option is C

$-\frac{i}{2}$

Explanation for the correct option.

Step 1. Find the value of $z^2$.

For the complex number $z=1+i$, the complex number $z^2$ is given as:

$\begin{array}{rcl}{z}^2& =& {\left(1+i\right)}^2\\ & =& 1+2i+i^2\\ & =& 1+2i+(-1)\left[i^2=-1\right]\\ & =& 1+2i-1\\ & =& 2i\end{array}$

Step 2. Find the multiplicative inverse of $z^2$.

The multiplicative inverse of a number $a$ is given as $\frac{1}{a}$.

So the multiplicative inverse of $z^2$ is $\frac{1}{z^2}$.

$\begin{array}{rcl}\frac{1}{z^2}& =& \frac{1}{2i}\left[z^2=2i\right]\\ & =& \frac{1·i}{2i·i}\\ & =& \frac{i}{2i^2}\\ & =& \frac{i}{2\times \left(-1\right)}\left[i^2=-1\right]\\ & =& -\frac{i}{2}\end{array}$

Thus, the multiplicative inverse of $z^2$ is $-\frac{i}{2}$.

Hence, the correct option is C.