Filling an array(Putnam)

Alan and Barbara play a game in which they take turns filling entries of an initially empty $ 2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

This question has been posted before.

I saw a solution recently this way:

When Alan places a number in any spot in the array, Barbara places the same number in the column, but one row up or down. That was she forces the rows to be linearly dependent, and the resulting determinate is zero.

This method works fine and I tried with arbitrary values in matrices of smaller rank.

But I need help in proving that this indeed forms linearly dependent rows which leads the determinant of the matrix to be zero?

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