How many $4 \times 4$ matrices with entries from $\{0, 1\}$ have odd determinant?
I was trying to partition the matrix as four block matrices with size $2 \times 2$, 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, please?
Best Answer
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.