If \(f: S \rightarrow \mathbb{R}\) where \(S\) is the set of all non-singular matrices of order 2 over \(\mathbb{R}\) and \(f\left[\begin{array}{ll}\left(\begin{array}{ll}0 & b \\ c & d\end{array}\right)\end{array}\right]=a d-b c\), then
(A) \(f\) is bijective mapping
(B) \(\mathrm{f}\) is one-one but not onto
(C) \(f\) is onto but not one-one
(D) \(f\) is neither one-one nor onto