How many 4 × 4 matrices with entries from {0, 1} have odd determinant? is there a short way of finding the answer to this question or do we have to solve it by hit and trial or using lengthy methods. if there is a short formula based answer to this,then please help me with this thank you
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This is the same as asking how many $4\times 4$ matrices with coefficients in $\mathbb{Z}/2\mathbb{Z}$ have nonzero determinant.
This is a standard argument: pick a nonzero vector for the first row, pick a vector not in the span of the first vector for the second row, etc. This gives us $(2^4-2^0) (2^4-2^1) (2^4-2^2) (2^4-2^3)$ possibilities.
Andrew Dudzik
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can u suggest me a book where i can find these kind of problem! which chapter should i go through? – Noel.campbell04091992 Sep 30 '14 at 13:48
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@Noel.campbell04091992 Depends on what you mean by "this kind of problem". I think it's important to get a solid background in linear algebra over general fields (and modules over rings, for that matter), which most abstract algebra texts have large sections devoted to—e.g. Dummit and Foote. – Andrew Dudzik Sep 30 '14 at 18:12