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How many 4 × 4 matrices with entries from {0, 1} have odd determinant? I was trying to partition the matrix as four block matrices with size 2 × 2, and consider all combinations of block matrices with determinants 0 and 1, such that determinant of the original matrix is odd. But I was stuck, as I was not sure about the relation between determinants of the block matrices and the original matrix. Can you help me out.

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up vote 3 down vote accepted

Hint: such matrices with even determinant have determinant zero in $\Bbb F_2$, whereas the ones with odd determinants have determinant 1 in $\Bbb F_2$ and hence are invertible. The question amounts to asking what the size of $GL(4,\Bbb F_2)$ is.

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