The inverse of $ z $ is:
$$ z^{-1} = \frac{ 3800 }{ 3293 }+\frac{ 4300 }{ 3293 }i $$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
$$ z_1 = \frac{1}{ \frac{ 19 }{ 50 }-\frac{ 43 }{ 100 }i } $$Step 2: Multiply top and bottom by complex conjugate of $ z $
$$ z_1 = \frac{1}{ \frac{ 19 }{ 50 }-\frac{ 43 }{ 100 }i } \cdot \frac{ \frac{ 19 }{ 50 }+\frac{ 43 }{ 100 }i }{ \frac{ 19 }{ 50 }+\frac{ 43 }{ 100 }i } $$Step 3: Simplify
$$ z_1 = \frac{ \frac{ 19 }{ 50 }+\frac{ 43 }{ 100 }i }{ \frac{ 3293 }{ 10000 } } $$$$ z_1 = \frac{ \frac{ 19 }{ 50 } }{ \frac{ 3293 }{ 10000 } } + \frac{ \frac{ 43 }{ 100 } }{ \frac{ 3293 }{ 10000 } } \cdot i$$$$ z_1 = \frac{ 3800 }{ 3293 }+\frac{ 4300 }{ 3293 }i $$