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Math Help - Determinant of 4 by 4 matrix

  1. #1
    Junior Member
    Joined
    Nov 2008
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    Determinant of 4 by 4 matrix

    Hi,
    I have a problem with understanding how to calculate a determinant of 4 by 4 matrix with no zero elements.
    If given matrix is:
    <br />
  A= \begin{pmatrix}a_{11} & a_{12} & a_{13} & a_{14}\\<br />
a_{21} & a_{22} & a_{23} & a_{24}\\a_{31} & a_{32} & a_{33} & a_{34}\\a_{41} & a_{42} & a_{43} & a_{44}\end{pmatrix}
    Find determinant.

    My try is:
    <br />
  det A= a_{11}  a_{22} a_{33} a_{44}- a_{11} a_{22} a_{34} a_{43}<br />
  -a_{11} a_{23} a_{32} a_{44} +a_{11} a_{23} a_{34} a_{42}

    <br />
   +a_{11} a_{24} a_{32} a_{43} -a_{11} a_{24} a_{33} a_{42}<br />
  -a_{12} a_{21} a_{33} a_{44} +a_{12} a_{21} a_{34} a_{43}<br />

    <br />
   +a_{12} a_{23} a_{31} a_{44} +a_{12} a_{23} a_{34} a_{41}<br />
   +a_{12} a_{24} a_{31} a_{43} -a_{12} a_{24} a_{33} a_{41}<br />

    <br />
  +a_{13} a_{21} a_{32} a_{44} -a_{13} a_{21} a_{34} a_{42} <br />
  -a_{13} a_{22} a_{31} a_{44} +a_{13} a_{22} a_{34} a_{41}

    <br />
   +a_{13} a_{24} a_{31} a_{42} -a_{13} a_{24} a_{32} a_{41} <br />
   -a_{14} a_{21} a_{32} a_{43} +a_{14} a_{21} a_{33} a_{42}

    <br />
   -a_{14} a_{22} a_{31} a_{43} -a_{14} a_{22} a_{33} a_{41}<br />
   +a_{14} a_{23} a_{31} a_{42} -a_{14} a_{23} a_{32} a_{41}<br />

    Is this a correct answer?
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  2. #2
    Super Member

    Joined
    May 2006
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    Lexington, MA (USA)
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    Hello, Bernice!

    I did not check your answer . . . too much work!


    A\;= \;\begin{pmatrix}a_{11} & a_{12} & a_{13} & a_{14}\\<br />
a_{21} & a_{22} & a_{23} & a_{24}\\a_{31} & a_{32} & a_{33} & a_{34}\\a_{41} & a_{42} & a_{43} & a_{44}\end{pmatrix} . . Find determinant.
    Can I assume you know the "Cofactor" method?


    \text{det }\!\!A \;=\;a_{11}\begin{vmatrix}a_{22}&a_{23}&a_{24} \\ a_{32}&a_{33}&a_{34} \\ a_{42}&a_{43}&a_{44} \end{vmatrix} - a_{12}\begin{vmatrix}a_{21}&a_{23}&a_{24}\\a_{31}&  a_{33}&a_{34}\\a_{41}&a_{43}&a_{44} \end{vmatrix} . +\:a_{13}\begin{vmatrix}a_{21}&a_{22}&a_{24}\\a_{3  1}&a_{32}&a_{34} \\ a_{41}&a_{42}&a_{44}\end{vmatrix} - a_{14}\begin{vmatrix}a_{21}& a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\a_{41}&a_{42}  &a_{43} \end{vmatrix}



    . . . = \;a_{11}\bigg( a_{22}\begin{vmatrix}a_{33}&a_{34}\\a_{43}&a_{44}\  end{vmatrix} - a_{23}\begin{vmatrix}a_{32}&a_{34}\\a_{42}&a_{44}\  end{vmatrix} + a_{24}\begin{vmatrix}a_{32}&a_{33}\\a_{42}&a_{43}\  end{vmatrix}<br />
\bigg)

    . . . . .  -a_{12}\bigg(a_{21}\begin{vmatrix}a_{33}&a_{34}\\a_  {43}&a_{44}\end{vmatrix} -a_{23}\begin{vmatrix}a_{31}&a_{34}\\a_{41}&a_{44}\  end{vmatrix} + a_{24}\begin{vmatrix}a_{31}&a_{33}\\a_{41}&a_{43} \end{vmatrix} \bigg)

    . . . . . . . +a_{13}\bigg(a_{21}\begin{vmatrix}a_{32}&a_{34}\\a  _{42}&a_{44}\end{vmatrix} -a_{22}\begin{vmatrix}a_{31}&a_{34}\\a_{41}&a_{44}\  end{vmatrix} + a_{24}\begin{vmatrix}a_{31}&a_{32}\\a_{41}&a_{42}\  end{vmatrix}\bigg)

    . . . . . . . . .  -a_{14}\bigg(a_{21}\begin{vmatrix}a_{32}&a_{33}\\a_  {42}&a_{43} \end{vmatrix} - a_{22}\begin{vmatrix}a_{31}&a_{33}\\a_{41}&a_{43}\  end{vmatrix} + a_{23}\begin{vmatrix}a_{31}&a_{32}\\a_{41}&a_{42}\  end{vmatrix} \bigg)


    . . . and so on.

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