Question

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

Answer 1

Ans : (D)
Hint : \(f\left[\left(\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right)\right]=4=f\left[\left(\begin{array}{ll}4 & 0 \\ 0 & 1\end{array}\right)\right]\)
\(\Rightarrow\) not one-one
As \(0 \in \mathbb{R}\) but \(S\) does not contain any singular matrix so, \(f\) is not onto

Answer 2

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Answer 3

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