Question

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

• A. $f$ is bijective mapping
• B. $f$ is one-one but not onto
• C. $f$ is onto but not one-one
• D. $f$ is neither one-one nor onto

Answer 1

The correct option is D $f$ is neither one-one nor onto

$f\left[\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\right]=ad-bcf\left[\left(\begin{array}{cc}2& 3\\ 4& 5\end{array}\right)\right]=10-12=-2f\left[\left(\begin{array}{cc}0& 1\\ 2& 0\end{array}\right)\right]=0-2=-2$
$\therefore$ Not one-one funtion
As the matrix is non singular matrix
So pre-image of zero doesn't exist
$\therefore$ Not onto