Matrix Tic Tac Toe

Rules
1. Player 0 places zeros in the matrix, Player 1 places ones.
2. Player 0 wins if the determinant of the final matrix is 0, Player 1 wins if it is nonzero.
Questions
Easy questions:
1. How can Player 0 know that she is a winner before the whole board is filled? (Think of at least two ways.)
2. Which player has the winning strategy on a 2x2 board if Player 0 starts?
3. Which player has the winning strategy on a 2x2 board if Player 1 starts?
4. Which numbers can we get as determinants on a 2x2 board?
Challenging questions:
1. Which numbers can we get as determinants on a 3x3 board?
2. Which player has the winning strategy on a 3x3 board if Player 0 starts?
3. Which player has the winning strategy on a 3x3 board if Player 1 starts?
Hard questions:
1. Which player has the winning strategy on a 4x4 or 5x5 board if Player 0 starts?
2. Which player has the winning strategy on a 4x4 or 5x5 board if Player 1 starts?

Determinant

Winner