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Thread: Determinant of 4 by 4 matrix

  1. #1
    Junior Member
    Joined
    Nov 2008
    Posts
    37

    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:
    $\displaystyle
    A= \begin{pmatrix}a_{11} & a_{12} & a_{13} & a_{14}\\
    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:
    $\displaystyle
    det A= a_{11} a_{22} a_{33} a_{44}- a_{11} a_{22} a_{34} a_{43}
    -a_{11} a_{23} a_{32} a_{44} +a_{11} a_{23} a_{34} a_{42}$

    $\displaystyle
    +a_{11} a_{24} a_{32} a_{43} -a_{11} a_{24} a_{33} a_{42}
    -a_{12} a_{21} a_{33} a_{44} +a_{12} a_{21} a_{34} a_{43}
    $

    $\displaystyle
    +a_{12} a_{23} a_{31} a_{44} +a_{12} a_{23} a_{34} a_{41}
    +a_{12} a_{24} a_{31} a_{43} -a_{12} a_{24} a_{33} a_{41}
    $

    $\displaystyle
    +a_{13} a_{21} a_{32} a_{44} -a_{13} a_{21} a_{34} a_{42}
    -a_{13} a_{22} a_{31} a_{44} +a_{13} a_{22} a_{34} a_{41}$

    $\displaystyle
    +a_{13} a_{24} a_{31} a_{42} -a_{13} a_{24} a_{32} a_{41}
    -a_{14} a_{21} a_{32} a_{43} +a_{14} a_{21} a_{33} a_{42}$

    $\displaystyle
    -a_{14} a_{22} a_{31} a_{43} -a_{14} a_{22} a_{33} a_{41}
    +a_{14} a_{23} a_{31} a_{42} -a_{14} a_{23} a_{32} a_{41}
    $

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

    Joined
    May 2006
    From
    Lexington, MA (USA)
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    Thanks
    848
    Hello, Bernice!

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


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


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



    . . . $\displaystyle = \;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}
    \bigg) $

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

    . . . . . . . $\displaystyle +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) $

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


    . . . and so on.

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