Let A = \left( {\matrix{ a & b \cr c & d \cr } } \right) where $a,b,c,d$ $\ne 0$ {A^2} = \left( {\matrix{ a & b \cr c & d \cr } } \right)\left( {\matrix{ a & b \cr c & d \cr } } \right) \Rightarrow {A^2} = \left( {\matrix{ {{a^2} + bc} & {ab + bd} \cr {ac + cd} & {bc + {d^2}} \cr } } \right) $\Rightarrow {a^2} + bc = 1,\,bc + {d^2} = 1$ $ab + bd = ac + cd = 0$ $c \ne 0\,\,\,\,\,b \ne 0$ $\Rightarrow a + d = 0 \Rightarrow Tr\left( A \right) = 0$ $\left| A \right| = ad - bc = - {a^2} - bc = - 1$