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Math Help - Determinants Question

  1. #1
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    Determinants Question

    Hi I ran across this problem and could not figure it out. Could someone please give me an explanation? I feel like I have something close but no cigar.
    Compute the following determinants by using row and/or column reduction


    \begin{bmatrix}3&-6&15\\-4&-5&-2\\2&7&3\end{bmatrix}


    Then, I applied the following row operations.
    R2->R2+2R3
    R3->3R3-2R1
    I also factored out the 3 from the top row.

    3 \begin{bmatrix}1&-2&5\\0&9&4\\0&33&-21\end{bmatrix}

    Then, R3->R3-33/9R2

    3  \begin{bmatrix}1&-2&5\\0&9&4\\0&0&\frac{-107}{3}\end{bmatrix}


    Det=3*1*9*-107/3=-963

    However, the answer should be -321.
    Last edited by egshih; July 30th 2009 at 11:43 PM. Reason: Forgot to include the directions
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  2. #2
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    Quote Originally Posted by egshih View Post
    Hi I ran across this problem and could not figure it out. Could someone please give me an explanation? I feel like I have something close but no cigar.
    Compute the following determinants by using row and/or column reduction


    \begin{vmatrix}3&-6&15\\-4&-5&-2\\2&7&3\end{vmatrix}


    Then, I applied the following row operations.
    R2->R2+2R3
    R3->3R3-2R1 Here's where it goes wrong. When you multiply R3 by 3 you multiply the determinant by 3. That's why you end up with 3 times the correct answer.
    I also factored out the 3 from the top row.

    3 \begin{vmatrix}1&-2&5\\0&9&4\\0&33&-21\end{vmatrix}

    Then, R3->R3-33/9R2

    3  \begin{vmatrix}1&-2&5\\0&9&4\\0&0&\frac{-107}{3}\end{vmatrix}


    Det=3*1*9*-107/3=-963

    However, the answer should be -321.
    Also, in LaTeX you should use vmatrix rather than bmatrix to get a determinant.
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  3. #3
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    \left| \begin{matrix}<br />
   \phantom{-}3 & -6 & \phantom{-}15  \\<br />
   -4 & -5 & -2  \\<br />
   \phantom{-}2 & \phantom{-}7 & \phantom{-}3 <br />
\end{matrix} \right|=3\left| \begin{matrix}<br />
   \phantom{-}1 & -2 & \phantom{-}5  \\<br />
   -4 & -5 & -2  \\<br />
   \phantom{-}2 & \phantom{-}7 & \phantom{-}3  <br />
\end{matrix} \right|, now the determinant equals 3\left| \begin{matrix}<br />
   1 & -2 & \phantom{-}5  \\<br />
   0 & \phantom{-}9 & \phantom{-}4  \\<br />
   0 & \phantom{-}11 & -7 <br />
\end{matrix} \right|=3\left| \begin{matrix}<br />
   9 & \phantom{-}4  \\<br />
   11 & -7 <br />
\end{matrix} \right|=-321.

    Don't waste your time producing more zeros.
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