Voltage Vs is applied at the input port with the output port short circuited

$V_1=Vs,\ V_2=0,\ I_2=-I_2'$

From Z parameter equations

$V_s=z_{11}I_1-z_{12}I_2^{'}$

$o=z_{21}I_1-z_{22}I_2^{'}$

$\therefore{} I_{1} {\Large = \frac{Z_{22}}{Z_{21}}\ \ I_2'}$

$\therefore{}Vs=Z_{11}\ \dfrac{Z_{22}}{Z21}\ I_2'-Z_{12}I_2'$

$\therefore{}\dfrac{Vs}{I_2'}=\dfrac{Z_{11}Z_{22}-Z_{12}Z_{21}}{Z_{21}}$

Voltage Vs is applied at the output port with input port short circuited

$V_2=Vs,\ V_1=0,\ {\ I}{1=\ -I'}\ $

From Z parameter equations

$O=\ -Z_{11}I_1'+Z_{12}I_2$ 

$\therefore{}I_2=\dfrac{Z_{11}}{Z_{12}}\ \ I_1$’

 $Vs=Z_{21}I_1^{'}+Z_{22}I_2$

$\therefore{}Vs=\ -Z_{21}I_1'+Z_{22}\dfrac{\ Z_{11}}{Z_{12}}\ \ I_1$’

$\therefore{}\dfrac{V_s}{I_1^{'}}=\dfrac{Z_{11}Z_{22}-Z_{21}Z_{12}}{Z_{12}}$

For the N/W to be reciprocal

$\dfrac{Vs}{I_1'}\ =\dfrac{Vs}{I_2'}$

$\therefore{} Z_{12}=Z_{21}$