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Linear algebra nxn matrix, n=14

  1. Feb 26, 2014 #1
    1. The problem statement, all variables and given/known data
    Use cofactor expansion to compute determinants of nxn matrices
    A= (aij)=
    [0 0 ... 0 1
    0 0 ... 2 0
    .............
    0 (n-1) 0 ... 0
    n 0 0 .... 0]

    B=(bij)=
    [ 0 1 0 ... 0
    0 0 2 ... 0
    .............
    0 0 0 ... (n-1)
    n 0 0 ... 0]


    2. Relevant equations

    det(A) = (aij)(-1^(i+j))det(aij)

    3. The attempt at a solution
    I though A was diagonal matrix so I tried (n)(n-1)2*1=364
    Then I realized to include ... so I thought det(A) = 0
    Both were wrong.
    Not sure how to solve either det(A) or det(B) when n=14
    Does than mean its just n=14 or is it 14x14 matrix?
     
  2. jcsd
  3. Feb 26, 2014 #2

    haruspex

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    You are on the right track, but you forgot one small thing. Try n = 2.
    How can it be the one without also being the other?
     
  4. Feb 26, 2014 #3
    1. if n= 2 then det(A) = 0
    2. Well if that is true then det(A) while n=14 should be zero, but I already tried that and the website for my homework says that is wrong.
     
  5. Feb 26, 2014 #4

    haruspex

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    No it isn't. What does A look like with n=2? How do you calculate the det of a 2x2 matrix?
     
  6. Feb 27, 2014 #5
    Well if n=2 (meaning a 2x2 matrix)
    A=
    [ 0 0
    0 0]
    and det(A) = ad-bc= 0*0-0*0=0 is this correct? Can you please explain how to solve with n=14?
     
  7. Feb 27, 2014 #6

    haruspex

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    No, you're misinterpreting the form of the matrix. For every n, the top row ends with a 1; for every n > 1, the second row ends with 2 0; the next, for n > 2, with 3 0 0; etc.
     
  8. Feb 27, 2014 #7
    So det(A) when n=14 is 14 factorial? I saw that pattern now and it looks like a diagonal matrix so det(A) should be product of the main diagonal entries.

    Or, if n=2 then det is is 2. If n=3 then det is 3*2. And if n=4 then det is 4*3*2?
    So if n=14, then det is 14!
     
  9. Feb 27, 2014 #8

    Dick

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    No, if n=2 det(A)=(-2). If n=3 det(A) is -3*2=(-6). If n=4 det(A) is +24. You have to explain the sign as well as the magnitude.
     
  10. Feb 28, 2014 #9
    so if n is 14, then det(A) = -1 (14!) ?
     
  11. Feb 28, 2014 #10

    Ray Vickson

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    What do *you* think? Of course, all answers must have some logical basis; what is yours?
     
  12. Feb 28, 2014 #11

    Dick

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    Why don't you try to use your "relevant equation" to answer that question?
     
  13. Feb 28, 2014 #12
    I think that the answer is -1*14!.
    Logical basis is the pattern that I can see from small n, that pattern goes 2*3*4*....*n with an alternating sign. Since n=14 occurs at row 14 column 1, sign is negative since -1^(14+1) is -1.
    Can you just give me a straight response, is this correct or not? I have only one attempt left for this problem and do not want to waste it. thanks
     
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