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Homework Help: Linear algebra Help!

  1. Feb 4, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the determinant of the matrix using identities without evaluating the determinants directly:

    a b b b
    b a b b
    b b a b
    b b b a

    3. The attempt at a solution
    I tried getting it into a triangular matrix but halfway through it got too complicated and it has to be simpler than what I think it is. The matrix is symmetric but I don't know how that relates to the determinant.
     
  2. jcsd
  3. Feb 4, 2010 #2
    I don't know if you are supposed to know this way but let's try :
    if a and b are real, a symmetric matrix is diagonalisable
    You know that the eigenvalues are roots of any polynom that cancel your matrix. For a start, compute the square of you matrix and reexpress it in terms of the indentity matrix and your original one. This identity gies you a polynom that cancel your matrix. Its roots are eigenvalues, and their product is the determinant. Take care of the multiplicity of the eigenvalues, that's all !

    You can also have guidance from basic aspects : you know trivial answers for special cases : a=b, a=0 or b=0 etc ... this should come then !

    Last possibility (but also not very elegant) : trying linear combination of lines and you will find a nice factorization appearing...
     
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