1
$\begingroup$

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?

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.