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Homework Help: One more

  1. Jan 1, 2007 #1
    just one last question on matrices, if you don't mind...

    Question:

    B is a 3*3 matrix det(B)= -3

    find det(B^T)

    (B^T is B transpose)

    My Answer:

    have none!

    help would be greatly appreciated
     
  2. jcsd
  3. Jan 1, 2007 #2
    What is the relationship between the 2 determinates we are looking at here? Does 'transposing' the matrix effect the determinate? if so, how?
     
  4. Jan 1, 2007 #3

    cristo

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    Well, to derive it, consider a general 3x3 matrix [tex] \left(\begin{array}{ccc}
    a&b&c\\d&e&f\\g&h&i \end{array}\right) [/tex] and expand the determinant

    [tex]
    \left|\begin{array}{ccc}
    a&b&c\\d&e&f\\g&h&i \end{array}\right|=
    a\left|\begin{array}{cc}e&f\\h&i\end{array}\right| - b\left|\begin{array}{cc}d&f\\g&i\end{array}\right|+c\left|\begin{array}{cc}d&e\\g&h\end{array}\right|=\cdots [/tex]

    Then consider the transposed matrix [tex] \left(\begin{array}{ccc}
    a&d&g\\b&e&h\\c&f&i \end{array}\right) [/tex] and expand this in a similar way. Compare the two results.
     
    Last edited: Jan 1, 2007
  5. Jan 1, 2007 #4
    it's the same!
    so the determinant of det(B^T) =det(B)=-3
     
  6. Jan 1, 2007 #5

    cristo

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    Correct. And in reply to your other thread, happy new year to you too!!
     
  7. Jan 1, 2007 #6
    thanx, my new years resolution is not to leave 200 questions till the last minute! it's nearly 2 am i'm going to finish off these ten questions...and go to sleep!
     
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