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Easy to see that these two determinants are identical

  1. Nov 11, 2013 #1
    1. The problem statement, all variables and given/known data
    Without expanding the determinant show that
    bc a^2 a^2
    b^2 ca b^2
    c^2 c^2 ab

    =
    bc ab ca
    ab ca bc
    ca bc ab


    2. Relevant equations



    3. Attempt at solution
    Well, one thing I noticed is that the diagonal row all contain the same values (bc, ca, ab)
    Using the first determinant, we can simplify it to
    bc |ca b^2| - ca|bc a^2| + ab |bc a^2|
    ....|c^2 ab|......|c^2 ab|.........|b^2 ca|

    the second determinant would be
    bc |ca bc| - ca |bc ca| + ab|bc ab|
    ....|bc ab|........|ca ab|........|ab ca|


    obviously its easy to see that these two determinants are identical, but is this what the question asks? It says to show without expanding, so I'm not sure if there's another way to show this.
     
    Last edited: Nov 11, 2013
  2. jcsd
  3. Nov 11, 2013 #2

    SteamKing

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    Your original determinant is somewhat garbled. You can insert text with
    Code (Text):
     tags to help preserve spacing.
     
  4. Nov 11, 2013 #3
    my bad, that should be easier to read now.
     
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