The inverse of $ z $ is:
$$ z^{-1} = -i $$We will denote the inverse of $ z $ as $ z_1 $. The inverse can be found in three steps.
Step 1: Rewrite complex number as its reciprocal
$$ z_1 = \frac{1}{ i } $$Step 2: Multiply top and bottom by complex conjugate of $ z $
$$ z_1 = \frac{1}{ i } \cdot \frac{ -i }{ -i } $$Step 3: Simplify
$$ z_1 = \frac{ -i }{ 1 } $$$$ z_1 = \frac{ 0 }{ 1 } - \frac{ 1 }{ 1 } \cdot i$$$$ z_1 = -i $$