Let S be the sample space of all 3x3 matrices with the entries from the set $\{0,1\}.$ Let the events  $E_{1}and{E}_2$ be given by  $E_1=\{A\in S:detA=0\}$ and $E_2=\{A\in S:sumoftheentriesofAis7\}$
If the matrix is chosen at random from S, then which of the following is correct?

Given sample space (S) of all 3 x 3 matrices with entries From the set  $\{0,1\}$ and events 
 $E_1=\{A\in S:\det \left(A\right)=0\}and E_2=\{A\in S:sumofentriesofAis7\}.$
For event  $E_2$, means sum of entries of matrix A is 7,
Then we need seven 1s and two 0s.
$\therefore$   Number of different possible matrices $=\frac{9!}{7!2!}$$\Rightarrow n\left(E_2\right)=36$ 
        
For event  $E_1$, |A| = 0, both the zeros must be in same row/column.
$\therefore$  Number of matrices such that their determinant is zero $=6\times \frac{3!}{2!}=18=n(E_1\cap E_2)$