Question

Let $M$ be the set of all $2\times 2$ matrices with entries from the set of real numbers $R$. Then the function $f:M\to R$ defined by $f\left(A\right)=\left|A\right|$ for every $A\in M$, is

• A. One-one and onto
• B. Neither one-one nor onto
• C. One-one but not onto
• D. Onto but not one-one

Answer 1

The correct option is D Onto but not one-one

$f:M\to R$ defined by $f\left(A\right)=\left|A\right|$ for every $A\in M$

The function is the determinant of the matrix

We know that, two different matrices can have a same determinant

For example, $A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$

Then $|A|=1=|B|$, but $A\ne B$

So, the function will not be one-one.

Now, determinants can have any real values, so range of the function will be $R$.

Thus, the function will be onto.

Hence, option D is correct