QJEEAdv19Maa852: JEE Advanced 2019 Mathematics - Class 12 JEEAdv19Maa852 Solution: Let S be the sample space of all 3 3 matrices with entries from …

Given sample space (S) of al 3 $\times$ 3 matrices with entries from the set {0, 1} and events

E₁ = {A $\in$ S : det(A) = 0} and

E₂ = {A $\in$ S : sum of entries of A is 7}.

For event E₂, means sum of entries of matrix A is 7, then we need seven 1s and two 0s.

$\therefore$ Number of different possible matrices = ${{9!} \over {7!2!}}$

$\Rightarrow$ n(E₂) = 36

For event E₁, |A| = 0, both the zeroes must be in same row/column.

$\therefore$ Number of matrices such that their determinant is zero = $6 \times {{3!} \over {2!}}$ = 18 = $n({E_1} \cap {E_2})$

$\therefore$ Required probability,

$P\left( {{{{E_1}} \over {{E_2}}}} \right) = {{n({E_1} \cap {E_2})} \over {n({E_2})}}$

$= {{18} \over {36}}$

$= {1 \over 2} = 0.50$