Let S be the sample space of all 3x3 matrices with the entries from the set ${0, 1} .$ Let the events $E_{1} a n d E_{2}$ be given by $E_{1} = {A \in S : d e t A = 0}$ and $E_{2} = {A \in S : s u m o f t h e e n t r i e s o f A i s 7}$
If the matrix is chosen at random from S, then which of the following is correct?

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$P (E_{1} \cap E_{2}) = \frac{1}{5}$

$P (E_{1} \cap E_{2}) = \frac{9}{256}$

$P (E_{1} \cap E_{2}) = \frac{7}{40}$

$P (E_{2}) = \frac{9}{128}$

answer is A, B.

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Detailed Solution

Given sample space (S) of all 3 x 3 matrices with entries From the set ${0, 1}$ and events
$E_{1} = {A \in S : \det (A) = 0} a n d E_{2} = {A \in S : s u m o f e n t r i e s o f A i s 7} .$
For event $E_{2}$ , means sum of entries of matrix A is 7,
Then we need seven 1s and two 0s.
$∴$ Number of different possible matrices $= \frac{9!}{7! 2!}$ $\Rightarrow n (E_{2}) = 36$

For event $E_{1}$ , |A| = 0, both the zeros must be in same row/column.
$∴$ Number of matrices such that their determinant is zero $= 6 \times \frac{3!}{2!} = 18 = n (E_{1} \cap E_{2})$