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Let S be the set of all 3 3 matrices having 3 entries equal to 1 and 6 entries equal to 0 . A matrix M is picked uniformly at random from the set S . Then the correct statement(s) is(are):

Options

  1. Atotal number of matrices in the set S is 84
  2. Bprobability that M is non-singular = 1 14
  3. Cprobability that M is identity matrix = 1 14
  4. Dprobability that M has trace equal to 0= 5 21

Correct answer

D. probability that M has trace equal to 0= 5 21

Step-by-step solution

Firstly, note that to calculate the total number of matrices in the sample space, we may place the three 1's in any of the 9 entries of M and the remaining 6 entries would be all 0 . Hence, total number of matrices M in the sample space is ^9 C₃=84 For M to be non-singular, all rows must be linearly independent so that M has full rank. Hence, each row must have exactly one 1 and no two 1 's must be present on the same column. This can be done in 6 ways. Hence, probability is 6 84 = 1 14 Prob (M=I₃ )= 1 84 because a

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