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2 Fixed a typo, r + s = n + p, not n - p.

Player A wins the trivial n=1 case by playing any non-zero number in (1,1). For all even n, player B wins by using symmetry a la a horizontal mirror. As Ben points out in the comments, if n = 3, player B can force a win. I had a long demonstration written out, but I decided against it (if you want, I can put it in later). Anyway, as for the general case, after a little searching, I found a paper called "A determinantal version of the Frobenius - König Theorem" by D. J. Hartfiel and Raphael Loewy, which can be purchased here.

The abstract, at least, says that given an n by n matrix A of, say, rational numbers, if the determinant is zero, then A must contain an r by s submatrix B such that r + s = n - + p, and rank(B) ≤ p - 1 (no more than p - 1 linearly independent rows), for some positive integer p. This means that if we have, say, a 5x5 matrix whose determinant is zero, then there exists a submatrix B in A such that B is:

1. a 1x5, 2x4, 3x3, 4x2, or 5x1 matrix of 0s
2. a 2x5, 3x4, 4x3, or 5x2 matrix whose rows are all scalar multiples of each other
3. a 3x5, 4x4, or 5x3 matrix with no more than two linearly independent rows
4. a 4x5 or 5x4 matrix with no more than three linearly independent rows
5. a 5x5 matrix with no more than four linearly independent rows (duh)

While it doesn't say so explicitly, I think that it's a biconditional, so if player B manages to get one of these in the matrix, then she will win. However, even if it isn't biconditional, if player A can prevent any of those forming, he will win.

Of these two, I believe it would be easier for player A to prevent any of these forming than it would be for player B to force one of these, but I haven't given that in particular a great deal of thought. I hope this is helpful.

1

Player A wins the trivial n=1 case by playing any non-zero number in (1,1). For all even n, player B wins by using symmetry a la a horizontal mirror. As Ben points out in the comments, if n = 3, player B can force a win. I had a long demonstration written out, but I decided against it (if you want, I can put it in later). Anyway, as for the general case, after a little searching, I found a paper called "A determinantal version of the Frobenius - König Theorem" by D. J. Hartfiel and Raphael Loewy, which can be purchased here.

The abstract, at least, says that given an n by n matrix A of, say, rational numbers, if the determinant is zero, then A must contain an r by s submatrix B such that r + s = n - p, and rank(B) ≤ p - 1 (no more than p - 1 linearly independent rows), for some positive integer p. This means that if we have, say, a 5x5 matrix whose determinant is zero, then there exists a submatrix B in A such that B is:

1. a 1x5, 2x4, 3x3, 4x2, or 5x1 matrix of 0s
2. a 2x5, 3x4, 4x3, or 5x2 matrix whose rows are all scalar multiples of each other
3. a 3x5, 4x4, or 5x3 matrix with no more than two linearly independent rows
4. a 4x5 or 5x4 matrix with no more than three linearly independent rows
5. a 5x5 matrix with no more than four linearly independent rows (duh)

While it doesn't say so explicitly, I think that it's a biconditional, so if player B manages to get one of these in the matrix, then she will win. However, even if it isn't biconditional, if player A can prevent any of those forming, he will win.

Of these two, I believe it would be easier for player A to prevent any of these forming than it would be for player B to force one of these, but I haven't given that in particular a great deal of thought. I hope this is helpful.