Question

Consider all $2\times 2$ matrices whose entries are distinct and belong to $\{1,2,3,4\}.$ The sum of determinants of all such matrices is

• A. is a negative integer
• B. $0$
• C. an odd positive integer
• D. $24$

Answer 1

The correct option is B $0$

Total number of possible matrices is $4!=24$
Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

and $B=\left[\begin{array}{cc}c& d\\ a& b\end{array}\right]$
Here, $B$ is obtained by interchanging the rows of $A.$
We can take $A$ and $B$ in pairs.
$\text{det}\left(A\right)=-\text{det}\left(B\right)$
$\Rightarrow \text{det}\left(A\right)+\text{det}\left(B\right)=0$