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Clarence

  • one year ago

I don't understand this question. Calculate the second term in the cofactor expansion along the third row for the matrix: [[-1,-1,-1,-1],[-1,1,2,0],[1,-1,0,2],[-1,-1,-1,-1]]

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  1. clarence
    • one year ago
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    \[\left[\begin{matrix}-1 & -1 & -1 & -1 \\ -1 & 1 & 2 & 0 \\ 1 & -1 & 0 & 2 \\ -1 & -1 & -1 & -1\end{matrix}\right]\]

  2. clarence
    • one year ago
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    So the third row would be [1, -1, 0, 2]

  3. clarence
    • one year ago
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    Second term being -1

  4. clarence
    • one year ago
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    So do I just use -1 times the rest of the matrix?

  5. ganeshie8
    • one year ago
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    \[\left[\begin{matrix}-1 & -1 & -1 & -1 \\ -1 & 1 & 2 & 0 \\ 1 & \color{red}{-1} & 0 & 2 \\ -1 & -1 & -1 & -1\end{matrix}\right]\]

  6. anonymous
    • one year ago
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    I guess you can take the determinant?

  7. anonymous
    • one year ago
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    Then multiply by -1

  8. clarence
    • one year ago
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    Take the determinant of the original matrix and times that by -1?

  9. ganeshie8
    • one year ago
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    whats the definition of "term" that your professor uses ?

  10. ganeshie8
    • one year ago
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    does the determinant has \(4!=24\) terms or just \(4\) terms according to your prof ?

  11. clarence
    • one year ago
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    I think she'd say that that'd be just 4 terms.

  12. ganeshie8
    • one year ago
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    ** then, you just need to find the cofactor of -1, which is at 3,2 location, then multiply that by -1

  13. ganeshie8
    • one year ago
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    ** cofactor of \(A_{32}\)= \((-1)^{3+2}\begin{vmatrix}-1&-1&-1\\-1&2&0\\-1&-1&-1\end{vmatrix}\)

  14. clarence
    • one year ago
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    Okay, I am obviously not understanding this question because the answer I got was 0...

  15. ganeshie8
    • one year ago
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    Right, I think your prof wants the first term out of those 24 terms in the determinant

  16. clarence
    • one year ago
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    There's a similar question on this website that went unanswered: http://openstudy.com/study#/updates/5039e2afe4b043c156a3277c

  17. clarence
    • one year ago
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    How did they obtain -20 in this example? https://au.answers.yahoo.com/question/index?qid=20120826032029AANjdNA

  18. ganeshie8
    • one year ago
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    the answer is either 0 or 2 0 if you consider 4 terms in the determinant 2 if you consider 24 terms

  19. ganeshie8
    • one year ago
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    yahoo answers is not that reliable..

  20. clarence
    • one year ago
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    Aha, I know, I was just trying to work out how they managed to get -20 that's all

  21. ganeshie8
    • one year ago
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    they got -40 right ?

  22. clarence
    • one year ago
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    Yeah.

  23. ganeshie8
    • one year ago
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    remember how to find cofactor of a term ?

  24. clarence
    • one year ago
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    Yes, but it's a bit hard to write it all on here.

  25. ganeshie8
    • one year ago
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    [3,-1,2,-1] [-2,2,1,2] [2,-2,2,1] [-1,3,2,-1] to find the cofactor of -2 at position 3,2 you simply find the determinant of the small matrix obtained by deleting 3rd row and second column, then fix the sign

  26. ganeshie8
    • one year ago
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    removing 3rd row and 2nd column, [3,2,-1] [-2,1,2] [-1,2,-1]

  27. clarence
    • one year ago
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    And then multiplying all that by -2.

  28. clarence
    • one year ago
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    What I don't understand is how they ended up getting (-2) x (-20) to get 40.

  29. ganeshie8
    • one year ago
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    before that, multiply the determinant by \((-1)^{3+2}\) to get the cofactor

  30. ganeshie8
    • one year ago
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    what do you get for determinant of [3,2,-1] [-2,1,2] [-1,2,-1]

  31. clarence
    • one year ago
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    Ohh... My bad..

  32. clarence
    • one year ago
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    So for my particular question, the determinant of the submatrix is just 0, does this mean that the answer is just 0 then?

  33. ganeshie8
    • one year ago
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    let me just quote my previous reply ``` the answer is either 0 or 2 0 if you consider 4 terms in the determinant 2 if you consider 24 terms ```

  34. clarence
    • one year ago
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    Yeah, I remember reading that, I was just confused as to how you managed to get 2 as well if I considered 24 terms rather than 4

  35. ganeshie8
    • one year ago
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    lets work it step by step

  36. ganeshie8
    • one year ago
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    how many terms are there in the determinant of a 2x2 matrix ?

  37. ganeshie8
    • one year ago
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    \[ \begin{vmatrix}a&b\\c&d\\\end{vmatrix} = ad-bc\] two terms, right ?

  38. clarence
    • one year ago
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    Yes.

  39. ganeshie8
    • one year ago
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    how about the determinant of a \(3\times 3\) matrix, how many terms ?

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