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Reducing a matrix determinant

  1. May 19, 2012 #1
    1. The problem statement, all variables and given/known data

    I've attached the problem, it involves reducing a 3x3 matrix determinant to row echelon form, but the leading diagonal elements have to be linear in a and b afterwards.

    2. Relevant equations


    3. The attempt at a solution

    I've managed to convert it to row echelon form by: r3-ar2 ; r2-ar1 ; r3-br2
    The problem is that this leaves a diagonal element having cubic terms. Can anyone see a way to acomplish this? Should be an easy problem, but I've spent over an hour trying different combinations.
     

    Attached Files:

  2. jcsd
  3. May 19, 2012 #2

    AKG

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    I recommend just computing the determinant and doing some simple algebra to factor the resulting expression.
     
  4. May 19, 2012 #3
    I also tried that, to no avail.
     
  5. May 19, 2012 #4

    Ray Vickson

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    The factorization is not immediately obvious, but what finally worked for me was to look at the determinant for two different numerical values of a (namely, a = 0 and a = 1) and in each case to factor the resulting polynomial in b. Some factors are the same for both values of a, and some others differ in such a way that you can easily figure out what they are as functions of a. You end up with a factorization exactly of the required type.

    RGV
     
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