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Math Help - determinant question 5

  1. #1
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    determinant question 5

    if \begin{vmatrix}<br />
a & b &c \\ <br />
d & e &f \\ <br />
 g&h  &k <br />
\end{vmatrix}=2

    then
    \begin{vmatrix}<br />
f & k-4c &2k+f \\ <br />
d & g-4a &2g+d \\ <br />
e&h-4b  &2h+e <br />
\end{vmatrix}=-16

    is it true?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by transgalactic View Post
    if \begin{vmatrix}<br />
a & b &c \\ <br />
d & e &f \\ <br />
 g&h  &k <br />
\end{vmatrix}=2

    then
    \begin{vmatrix}<br />
f & k-4c &2k+f \\ <br />
d & g-4a &2g+d \\ <br />
e&h-4b  &2h+e <br />
\end{vmatrix}=-16

    is it true?
    I don't know, is it?
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  3. #3
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    i dont know how to break the big determinant into smaller packages of the original determinant
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  4. #4
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    Have you tried computing the determinants for both and comparing them? Although I'm not completely sure if this is the best way, but it may work.
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  5. #5
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    Quote Originally Posted by Drexel28 View Post
    I don't know, is it?

    Remember that determinant is a multilineal functions, so:

    \begin{vmatrix}f & k-4c &2k+f \\ d & g-4a &2g+d \\ e&h-4b &2h+e \end{vmatrix}=\begin{vmatrix}f & k &2k \\ d & g &2g \\ e&h &2h \end{vmatrix}+\begin{vmatrix}f & k &f \\ d & g &d \\ e&h &e \end{vmatrix}+ \begin{vmatrix}f & -4c &2k \\ d & -4a &2g \\ e&-4b &2h \end{vmatrix}+\begin{vmatrix}f & -4c &f \\ d & -4a &d \\ e&-4b &e \end{vmatrix} ,

    So....??

    Tonio
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tonio View Post
    Remember that determinant is a multilineal functions, so:

    \begin{vmatrix}f & k-4c &2k+f \\ d & g-4a &2g+d \\ e&h-4b &2h+e \end{vmatrix}=\begin{vmatrix}f & k &2k \\ d & g &2g \\ e&h &2h \end{vmatrix}+\begin{vmatrix}f & k &f \\ d & g &d \\ e&h &e \end{vmatrix}+ \begin{vmatrix}f & -4c &2k \\ d & -4a &2g \\ e&-4b &2h \end{vmatrix}+\begin{vmatrix}f & -4c &f \\ d & -4a &d \\ e&-4b &e \end{vmatrix} ,

    So....??

    Tonio
    Why did you quote me? I know how to do this.
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  7. #7
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    Quote Originally Posted by Drexel28 View Post
    Why did you quote me? I know how to do this.

    I'm a selective reader: only got "I don't know" and thought it was the OP. Hopefully I won't confuse him/her.

    Tonio
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  8. #8
    MHF Contributor matheagle's Avatar
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    I'd do column reduction and then factor out the 2 and -4, so 2(2)(-4)=-16
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