${A^2} = \left[ {\begin{array}{*{20}{c}}
a&b\\
c&d
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
a&b\\
c&d
\end{array}} \right]$

$ = \left[ {\begin{array}{*{20}{c}}
{{a^2} + bc}&{ab + bd}\\
{ac + cd}&{bc + {d^2}}
\end{array}} \right] = I$

${a^2} + bc = bc + {d^2} = 1$

$ac + cd = ab + bd = 0$

$ac + cd = ab + bd = 0$

$b\left( {a + d} \right) = 0$

$c = 0\;\:{\rm{or}}\;\:a =  - d$ not possible for $c$

$b = 0\;\:{\rm{or}}\;\:a =  - d$ not possible for $b$

$\left| {\begin{array}{*{20}{c}}
a&b\\
c&d
\end{array}} \right| = ad - bc =  - {d^2} - bc$

$ =  - \left( {{d^2} + bc} \right) =  - 1$

$tr\left( A \right) = a + d = a - a = 0$