Let $$A = \left( {\matrix{ a & b \cr c & d \cr } } \right)$$ be a 2 $$ \times $$ 2 real matrix with det A = 1. If the equation det (A $$ - $$ $$\lambda $$I₂) = 0 has imaginary roots (I₂ be the identity matrix of order 2), then

If $$\left| {\matrix{ {{a^2}} & {bc} & {{c^2} + ac} \cr {{a^2} + ab} & {{b^2}} & {ca} \cr {ab} & {{b^2} + bc} & {{c^2}} \cr } } \right| = k{a^2}{b^2}{c^2}$$,

then K =

If f : S $$ \to $$ R, where S is the set of all non-singular matrices of order 2 over R and $$f\left[ {\left( {\matrix{ a & b \cr c & d \cr } } \right)} \right] = ad - bc$$, then

If the vectors $$\alpha = \widehat i + a\widehat j + {a^2}\widehat k,\,\beta = \widehat i + b\widehat j + {b^2}\widehat k$$ and $$\,\gamma = \widehat i + c\widehat j + {c^2}\widehat k$$ are three non-coplanar

vectors and $$\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr } } \right| = 0$$, then the value of abc is