Players A en B play a game. They take an empty n-by-n matrix (n > 0) and place one by one an element (say a rational number) in an unoccupied place of this matrix. Player A starts. The game ends if there is no move left. Player A wins if the matrix is invertible, player B wins if it is not. Is there, for a given n, a winning strategy for one of the two players?

It is not hard to show that for n = 3, player A can win. Also if n is even player B has a winning strategy. But what if n is odd and n > 3?