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Linear Algebra, scalars to find det(A)

  1. Feb 25, 2014 #1
    1. The problem statement, all variables and given/known data
    Let X1, X2,.....,Xn be scalars. Calculate det(A) where
    A= nxn matrix with [ x1+1 x1+2......x1+n
    x2+1 x2+2......x2+n
    .... .... .....
    xn+1 xn+2......xn+n]



    2. Relevant equations
    det(A) = (aij)(-1)^(i+j)det(aij)



    3. The attempt at a solution

    I have no clue how to even begin solving this problem I tried using above formula to no avail.
    Please help!
     
  2. jcsd
  3. Feb 25, 2014 #2
    Spacing on matrix got messed up so imagine it with the rows lined up correctly
     
  4. Feb 25, 2014 #3

    AlephZero

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    Review what you know about how determinants, row (or column) operations on the matrix, and linear dependence.

    Hint: the answer is very simple, when n > 2.
     
  5. Feb 25, 2014 #4
    Can you explain a little more about what operations you are talking about? I am really confused!
     
  6. Feb 25, 2014 #5
    Have you tried computing the determinant for small n? Such as a 2x2, 3x3, 4x4. I'd recommend doing that.
     
  7. Feb 25, 2014 #6
    How would that help me calculate the determinant when n is unknown?
     
  8. Feb 25, 2014 #7

    Ray Vickson

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    Start with simple cases of, say, n = 2, 3 and maybe 4, just to see what is going on. Then do it for general n.
     
  9. Feb 25, 2014 #8

    AlephZero

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    The answer doesn't depend on n, except that n = 1 and n = 2 are special cases.

    I don't know how to give you any more help than "think about row and column operations on the matrix, and linear dependence" without telling you the answer.

    Writing out the matrix in full, for n = 3 or n = 4, might help.

    If the only thing you know about determinants is your "relevant equation"
    det(A) = (aij)(-1)^(i+j)det(aij)
    I think you missed a lot of stuff in class, or you haven't read your textbook.
     
  10. Feb 26, 2014 #9
    Yes I did computr for 2 by 2 and 3 by 3, but I do not see a relationship between the determinants. How do I calculate det(A) for nxn?
     
  11. Feb 26, 2014 #10
    Okay I did determinant calculation for 2x2 and 3x3. How do I calculate for nxn? Please help!
     
  12. Feb 26, 2014 #11

    pasmith

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    Two hints:
    (1) Why is [itex]\det A = B(x_2, \dots, x_n)x_1 + C(x_2, \dots, x_n)[/itex] for some functions [itex]B[/itex] and [itex]C[/itex]?
    (2) What is the determinant of a matrix with two identical rows?
     
  13. Feb 26, 2014 #12
    1. I have no clue

    2. determinant would be zero. Is the answer zero?
     
  14. Feb 26, 2014 #13

    Ray Vickson

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    So, what did you get for n = 2 and for n = 3?
     
  15. Feb 27, 2014 #14
    For n=2 det(A)= (x1 +1)(x2 +2) - (x1 +2)(x2 +1)
    How does that correlate to unknown n?
     
  16. Feb 27, 2014 #15

    Ray Vickson

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    What do you get for n = 3? I mean, expand out everything and simplify it down to as small an expression as you can get. I am 100% serious. Looking at just n = 2 is not enough to reveal the pattern.
     
  17. Feb 27, 2014 #16

    AlephZero

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    Can you think of a way to do row or column operations on the matrix, to make two rows identical, and not change the determinant?
     
  18. Feb 27, 2014 #17
    Hey I actually just figured out how to solve it by using row operations and linear dependence and I got zero as the determinant. Is this correct?
     
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