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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?
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.
whenever 1st row is 0 then its determent is 0 , and similarly if any 2 or more rows are linearly dependent then its |det|=0
In order to find the odd determinant the 1st row must be non zero ⇒ totally(2^4 −1) possibilities
2nd row must be non zero and not linearly depends on 1st row so⇒ totally (2^4 −2) possibilities
for 3rd row it must be non-zero as well as not linearly depends on first 2 rows(not start with 0) ⇒ totally (2^4 −4)
for 4th row ⇒(2^4 −8)
:: total possibilities=(2^4 − 2^0)∗(2^4 − 2^1)∗(2^4 − 2^2)∗(2^4 −2^3) = 15∗14∗12∗8=20160 possible
