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Weird failure of Cramers Rule

  1. Aug 10, 2012 #1
    This is very weird, but I found an inconsistency in the application of Cramer's Rule for a 3x3 simple linear matrix.

    1x + 1y + 0z = 3
    -1x + 3y + 4z = -3
    0x + 4y + 3z = 2

    Dz =
    1 1 3
    -1 3 -3
    0 4 2

    If you take the determinant across the first row To find Dz, I constantly get -16

    If you take the determinant across any other rows or columns, you get the correct Dz = 8

    What is going on?????????

    Help please.
  2. jcsd
  3. Aug 10, 2012 #2
    Not really going to be able to help without seeing a step by step calculation. I get 8 no matter which minor I choose to expand by.
  4. Aug 10, 2012 #3
    Omg! Sorry... I calculated incorrectly by forgetting a negative.

    My mistake was


    It should have been

  5. Aug 10, 2012 #4

    What do you mean by "to take the determinant across a row? Do you mean to calculate it wrt the
    minors determined by that row? Let's see:
    [tex]\left|\begin{array}{rrr}1&1&3\\-1&3&-3\\0&4&2\end{array}\right|= 1\cdot\left|\begin{array}{rr}\,3&-3\\\,4&2\end{array}\right|+(-1)\cdot\left|\begin{array}{rr}-1&-3\\0&2\end{array}\right|+3\cdot\left|\begin{array}{rr}-1&3\\0&4\end{array}\right|=(6+12)-(-2)+3(-4)=18+2-12=8[/tex]

    If you meant the above then the result is 8, which is hardly surprising as this is the matrix's determinant ; if you

    meant something else then I can't say.

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