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Determinant proof

  1. Dec 10, 2007 #1

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    Last edited: Dec 10, 2007
  2. jcsd
  3. Dec 10, 2007 #2

    Dick

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    The determinant is a multilinear function of the columns. That means every term in the polynomial expansion of the determinant contains exactly one entry from the first column. How's that for a direction? Think about expanding by minors.
     
  4. Dec 11, 2007 #3
    Im still lost. Im not understanding.
     
  5. Dec 11, 2007 #4

    robphy

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    A related problem (actually a special case)... concerning cross-products involving vectors A, B, and M.


    (A x M) + (B x M) = (A+B) x M

    Can you do the 2x2 version of your question? The 1x1 is easy.
     
  6. Dec 11, 2007 #5

    HallsofIvy

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    Do you know how to "expand by minors"? If so find the determinant on the right by expanding along the first column.
     
  7. Dec 11, 2007 #6

    Dick

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    Look up 'expansion by minors'. You can turn the first 3x3 determinant into a sum of three 2x2 determinants a_11*A_11-a_21*A_21+a_31*A_31, where A_ij is a 2x2 determinant that doesn't include any elements of the first column. The second one is b_11*A_11-b_21*A_21+b_31*A_31. The important thing is that the A determinants are the same.
     
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