The inverse of $ z $ is:
$$ z^{-1} = \frac{ 1 }{ 5 }+\frac{ 2 }{ 5 }i $$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
$$ z_1 = \frac{1}{ 1-2i } $$Step 2: Multiply top and bottom by complex conjugate of $ z $
$$ z_1 = \frac{1}{ 1-2i } \cdot \frac{ 1+2i }{ 1+2i } $$Step 3: Simplify
$$ z_1 = \frac{ 1+2i }{ 5 } $$$$ z_1 = \frac{ 1 }{ 5 } + \frac{ 2 }{ 5 } \cdot i$$$$ z_1 = \frac{ 1 }{ 5 }+\frac{ 2 }{ 5 }i $$