Let S be the sample space of all 3×3 matrices with entries from the set {0, 1}. Let the events E₁ = {A∈S : det A = 0} and E₂ = {A∈S : Sum of entries of A is 7}. If a matrix is chosen at random from S, then the conditional probability P(E₁|E₂) equals.

Correct option is A. 0.50
E₂: Sum of elements of A = 7 ⇒ These are 7 ones and 2 zeroes
Number of such matrices = ⁹C₂ = 36.
Out of all such matrices; E₁ will be those when both zeros lie in the same row or in the same column
e.g. [111
011
011]
n(E₁∩E₂) = 2 × ³C₂ × ³C₂ = 18
So n(E₁/E₂) = n(E₁∩E₂)/n(E₂) = 18/36 = 1/2.