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1. The problem statement, all variables and given/known data

The elements of a determinant are arbitrary integers. Determine the probability that the value of the determinant is odd. (Hint: Work mod 2).

2. Relevant equations

3. The attempt at a solution

When n=1, the probability is 1/2. When n = 2, the probability is 3/8. That is the last case I can do by hand. I am trying to find a recursion, but that is hard as the following example illustrates.

For n=3, if you expand along the first column, you can multiply the probability that a_{11} is odd by the probability that its cofactor is odd and than do the same for a_{21} and a_{31}. You can ignore the sign of the cofactor because we are working mod 2. So, we can just add these three terms.

It would be nice if you could apply the n=2 case to the cofactors, but the probability the cofactor of a_{11} is odd is not independent of the probability that the a_{21} cofactor is odd, so I don't think that you can just replace those probabilities by P_{n-1}.

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# Homework Help: Probability a determinant is odd

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