How Sure Can You Be?
Sayan wishes to calculate the inverse of a 3x3 matrix using adjoint method. However, there is a 10% chance that he calculates the cofactor of an element incorrectly.
After writing the cofactors' matrix, Sayan feels uncertain and randomly chooses an element and checks its cofactor. He sees that he has calculated the cofactor correctly and feels relieved. Let the probability that he has in fact computed the adjoint correctly be where and are coprime natural numbers. What is the value of ?
(Assumptions: The checking is done with 100% acccuracy.
He remembers to take the transpose of the cofactors' matrix.)
(Bonus Question: What if the checking is not done with 100% accuracy and is done with, say, 95% accuracy?)
The answer is 143046721.
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Given that one cofactor is correct, the probability that the rest are correct is ( 1 0 9 ) 8 = 1 0 0 0 0 0 0 0 0 4 3 0 4 6 7 2 1 . This gives us an answer of 4 3 0 4 6 7 2 1 + 1 0 0 0 0 0 0 0 0 = 1 4 3 0 4 6 7 2 1 .