Question

$M$ is the set of all $2\times 2$ real matrices. $f:M\to R$ is defined by $f\left(A\right)=detA$ for all $A$ in $M$ then $f$ is

• A. one-one but not onto
• B. onto but not one-one
• C. neither one-one nor onto
• D. bijective

Answer 1

The correct option is B onto but not one-one

$f\left(\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\right)=f\left(\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\right)=1\Rightarrow f$ is not one-one

$\forall K\in R$, there exist a matrix A $=\left(\begin{array}{cc}1& 0\\ 0& k\end{array}\right)$
such that $f\left(A\right)=k\Rightarrow f$ is onto.
Ans: B