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If $\mathrm{f}: \mathrm{S} \rightarrow \mathbb{R}$ where $S$ is the set of all non-singular matrices of order 2 over $\mathbb{R}$ and $f\left[\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)\right]=a d-b c$, then
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Verified Answer
The correct answer is:
$\mathrm{f}$ is neither one-one nor onto
Hint :
$\mathrm{f}\left[\left(\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right)\right]=4=\mathrm{f}\left[\left(\begin{array}{ll}4 & 0 \\ 0 & 1\end{array}\right)\right]$
$\Rightarrow$ not one-one
As $0 \in \mathbb{R}$ but S does not contain any singular matrix so, $f$ is not onto
$\mathrm{f}\left[\left(\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right)\right]=4=\mathrm{f}\left[\left(\begin{array}{ll}4 & 0 \\ 0 & 1\end{array}\right)\right]$
$\Rightarrow$ not one-one
As $0 \in \mathbb{R}$ but S does not contain any singular matrix so, $f$ is not onto
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