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?
1. **Initial Setup**: Alan and Barbara are playing a game on a \(2008 \times 2008\) matrix. Alan wins if the determinant of the final matrix is nonzero, and Barbara wins if the determinant is zero. Alan plays first.

2. **Barbara's Strategy**: To ensure that the determinant of the matrix is zero, Barbara can use the following strategy:
   - Whenever Alan places a number in any cell of the matrix, Barbara will place a zero in the corresponding cell of another row or column, effectively making two rows or two columns identical.

3. **Detailed Strategy**:
   - Suppose Alan places a number \(a_{ij}\) in the cell at row \(i\) and column \(j\).
   - Barbara will then place a zero in the cell at row \(i+1\) (or any other row that hasn't been used yet) and column \(j\).
   - By continuing this strategy, Barbara ensures that at least two rows (or columns) of the matrix will be identical.

4. **Determinant Calculation**:
   - If two rows (or columns) of a matrix are identical, the determinant of the matrix is zero. This is a fundamental property of determinants.
   - Therefore, by making sure that at least two rows (or columns) are identical, Barbara guarantees that the determinant of the matrix will be zero.

5. **Conclusion**:
   - Since Barbara can always place a zero in a way that makes two rows or columns identical, she has a winning strategy.
   - Alan cannot prevent Barbara from making the determinant zero, regardless of his moves.

\[
\boxed{\text{Barbara has a winning strategy.}}
\]