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How can I make this matrix a triangular one?

  1. Apr 29, 2012 #1
    Hi. How can I reduce this matrix into a triangular one so I can calculate the determinant easily.

    [tex]\displaystyle\left( {\begin{array}{*{20}{c}}
    b&1&1&1&1 \\
    1&b&1&1&1 \\
    1&1&b&1&1 \\
    1&1&1&b&1 \\
    1&1&1&1&b
    \end{array}} \right)[/tex]

    I've tried but I cannot make a triangular form...

    Thanks!
     
  2. jcsd
  3. Apr 29, 2012 #2

    sharks

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    Reduce to row echelon form by Gaussian elimination.
     
  4. Apr 29, 2012 #3

    Hurkyl

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    Have you tried computing the determinant by other means?
     
  5. Apr 29, 2012 #4

    sharks

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    If you need to find the determinant, try the cofactor expansion by the first row.
     
  6. Apr 29, 2012 #5
    I could! I subtracted row 1 from all rows and then I subtracted the all columns from the first.
     
  7. Apr 29, 2012 #6

    Hurkyl

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    The cofactor expansion was the one I was thinking of earlier.

    Yet another approach is to write the matrix as the sum A + (b-1) I, where A is the matrix of all 1's, and I is the identity matrix. A is diagonalizable, and you can find its eigenvalues without too much trouble -- and so you should also be able to find the eigenvalues of the sum!
     
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